EXISTENCE OF HYDRODYNAMICS FOR THE ... - Project Euclid

The Annals of Probability 1999, Vol. 27, No. 1, 361 ᎐ 415

EXISTENCE OF HYDRODYNAMICS FOR THE TOTALLY ASYMMETRIC SIMPLE K-EXCLUSION PROCESS BY TIMO SEPPALAINEN ¨¨ Iowa State University In a totally asymmetric simple K-exclusion process, particles take nearest-neighbor steps to the right on the lattice Z, under the constraint that each site contain at most K particles. We prove that such processes satisfy hydrodynamic limits under Euler scaling, and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function. Our technique requires no knowledge of the invariant measures of the process, which is essential because the equilibria of asymmetric K-exclusion are unknown. But we cannot calculate the flux function precisely. The proof proceeds via a coupling with a growth model on the two-dimensional lattice. In addition to the basic K-exclusion with constant exponential jump rates, we treat the sitedisordered case where each site has its own jump rate, randomly chosen but frozen for all time. The hydrodynamic limit under site disorder is new even for the simple exclusion process Žthe case K s 1.. Our proof makes no use of the Markov property, so at the end of the paper we indicate how to treat the case with arbitrary waiting times.

1. Introduction. The family of simple K-exclusion processes naturally interpolates between the simple exclusion process ŽSEP. and the zero-range process ŽZRP.. In K-exclusion the particles interact through the K-exclusion rule that stipulates that each site contain at most K particles. The case K s 1 is the SEP. The ZRP has no such restriction, so it can be thought of as the case K s ⬁. The SEP and ZRP are among the most fruitful models for studies of hydrodynamic behavior of interacting particle systems. By contrast, the K-exclusion processes have been harder to work with. The difficulty posed by the symmetric K-exclusion is that it violates the gradient condition w see II.2.4 in Spohn Ž1991.x . On the positive side, there are reversible product measures that can be easily identified. With Varadhan’s Ž1993. nongradient techniques, the diffusive hydrodynamic limit for symmetric K-exclusion can be proved, as was demonstrated by Kipnis, Landim and Olla Ž1994.. In this paper we address the hydrodynamics of totally asymmetric Kexclusion. In the asymmetric case, invariant measures are unknown, and even the belief that there is a spatially ergodic equilibrium for each density remains unproved. We prove the existence of a hydrodynamic limit for the empirical particle density and a qualitative characterization of the limiting

Received May 1998. AMS 1991 subject classifications. Primary 60K35; secondary 60K25, 82C22, 82C24. Key words and phrases. K-exclusion process, marching soldiers model, MrMr1rK queues in series, quenched disorder, hydrodynamic limit, subadditive process, growth model.

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¨ ¨ T. SEPPALAINEN

macroscopic density. As expected, it is the unique entropy solution of a scalar conservation law with a concave flux function. However, without knowledge of the equilibria, or without some other new idea, we cannot explicitly express the flux f K Ž ␳ . as a function of density ␳ . We can bound f K with the explicitly known flux functions of the cases K s 1 and K s ⬁. We also prove the hydrodynamic limit for the site-disordered totally asymmetric K-exclusion. Now the jump rates of the sites are i.i.d. random variables, picked once and fixed for the duration of the dynamics. This situation goes by the term ‘‘quenched disorder.’’ This result is new even for the case K s 1, the totally asymmetric simple exclusion process ŽTASEP.. The case K s ⬁, a ZRP with site disorder, is more accessible because it has product-form equilibria that can be written down explicitly. The hydrodynamics of the site-disordered asymmetric ZRP, or equivalently, the particle-disordered SEP, have been studied by Benjamini, Ferrari and Landim Ž1996., and Seppalainen ¨¨ and Krug Ž1998.. The theorems are stated and proved in the continuous-time setting. Without essential changes, the same proofs work for the discrete-time K-exclusion process where the entire configuration is updated simultaneously at each time step. One just replaces exponential waiting times with geometric ones and Poisson point processes of jump times with i.i.d. processes of points on the positive integers. The approach also extends to non-Markovian processes, but here new proofs are needed part of the way. The non-Markovian K-exclusion operates with i.i.d. waiting times that have an arbitrary common distribution on Ž0, ⬁.. The key to our proof is a coupling of a process with an arbitrary initial configuration with a family of processes with simple initial configurations. This coupling gives a variational equation that relates the general process to the simple ones. The hydrodynamics of the simple processes can be handled by Kingman’s Ž1968. subadditive ergodic theorem. To achieve this, the simple process is recast as a growth model on the planar lattice. This growth model in turn is formulated in terms of ‘‘ vertex greedy lattice paths’’ w Cox, Gandolfi, Griffin and Kesten Ž1993.x that satisfy certain constraints on the admissible steps of the path. Along the way we obtain a result of interest for the growth model, namely the Legendre duality of the macroscopic shape and the macroscopic velocity. The macroscopic shape is a function defined on a reference line, which in our model is the y-axis, and the macroscopic velocity is a function of the local slope. See Section 2 in Krug and Spohn Ž1991.. Another process closely related to the K-exclusion is the marching soldiers model. ŽThe increments of the marching soldiers model can be transformed into a K-exclusion.. As a corollary of our proofs, we obtain a hydrodynamic limit for totally asymmetric marching soldiers as well. The strength of our approach is that it gives sharp results with minimal assumptions. For example, for the hydrodynamic limits we need no extra assumptions on the initial distributions of the processes, only the minimal assumption that a macroscopic profile exists. Furthermore, we get not

HYDRODYNAMICS FOR K-EXCLUSION

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only weak laws of large numbers, but also strong laws. The disadvantage is that the technique relies on the existence of a special structure, namely, the coupling with simple processes mentioned above. Other applications of related ideas to hydrodynamic limits, asymptotic shapes, and large deviations Ž1996, 1997a, b, 1998a, b, appear in Aldous and Diaconis Ž1995., Seppalainen ¨¨ c. and Seppalainen and Krug Ž1998.. ¨¨ Further relevant literature. For general accounts of particle systems and their construction with Poisson point processes of jump times we refer the reader to Durrett Ž1988., Griffeath Ž1979. and Liggett Ž1985.. The equilibria of the zero-range process were characterized by Andjel Ž1982.. For hydrodynamical limits there are lectures by De Masi and Presutti Ž1991., the monograph of Spohn Ž1991. and review papers by Ferrari Ž1994, 1996., together with their references. For ZRP and SEP, the results that correspond to our theorems were proved by Rezakhanlou Ž1991.. The approach of his paper was extended to some inhomogeneous models by Landim Ž1996. and Covert and Rezakhanlou Ž1997.. Organization of the paper. The reader who wants a quick look at the central ideas of the paper should set ␣ Ž j . s 1 everywhere, read Section 2 up to Theorem 1, Section 4, Section 5 up to Corollary 5.1, and Section 6. The key to the proof of the hydrodynamic limit is equation Ž6.5. in Section 6, to which one applies assumption Ž2.4. w in the form Ž6.3.x and Corollary 5.1. The results for K-exclusion and the marching soldiers model, without disorder, are stated and discussed in Section 2. In this setting we prove both a weak law and a strong law for the empirical particle density. A weak law for disordered K-exclusion is stated in Section 3. The coupling that is central to the proof is developed in Section 4. The growth model is defined and its scaling limit proved in Section 5. The law of large numbers for the interface of the growth model drives the hydrodynamic limits, through the coupling explained in Section 4. In Section 6 the weak laws stated in Sections 2 and 3 are proved. The passage from weak law to strong law goes through a Borel᎐Cantelli argument. Section 7 is devoted to a summable upper tail probability estimate for the passage times of the growth model. Armed with this estimate, Section 8 proves the strong laws stated in Section 2. Section 9 develops some bounds for the limiting shape of the growth model and for the flux function of the conservation law of the particle density. Section 10 treats the non-Markovian case. We outline the proof of the weak law of the hydrodynamic limit. With additional assumptions on the waiting times, the strong law proof generalizes as well. Notational remarks. N is the set 1, 2, 3, . . . 4 of natural numbers, while Zqs 0, 1, 2, 3, . . . 4 . A nonnegative random variable Y is ExpŽ ␤ .-distributed if P Ž Y ) t . s ey␤ t for all t ) 0, w x x s max n g Z: n F x 4 for x g R. IA and I A4 denote the indicator random variable of the event A.


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2. Totally asymmetric simple K-exclusion. The process we study has the following informal description. Sites on the lattice Z are occupied by indistinguishable particles. Each site contains at most K particles. K is a positive integer constant whose value is arbitrary, but fixed. Each site i g Z has an independent rate 1 Poisson point process Di on the time line 0 F t - ⬁. The epochs of the Poisson processes are potential jump times. At each epoch of Di , this event takes place: if there is at least one particle at i and at most K y 1 particles at i q 1, one particle from site i jumps to site i q 1. In other words, jumps are executed as long as there is a particle to jump and as long as the K-exclusion rule is not violated. All jumps proceed to the right, hence the modifier ‘‘totally asymmetric’’ in the name of the process. The modifier ‘‘simple’’ refers to the restriction that only nearest-neighbor jumps are allowed. The state of the process is a configuration ␩ s Ž␩ Ž i .: i g Z. of occupation numbers, where ␩ Ž i . g 0, 1, . . . , K 4 specifies the number of particles at site i. The compact state space of the process is X s 0, 1, . . . , K 4 Z . The dynamics can be conveniently represented by the generator L of the process that acts on cylinder functions f on X:

Ž 2.1.

Lf Ž ␩ . s

Ý I␩ Ž i.G 1 , ␩ Ž iq1.F Ky14 f Ž␩ i , iq1 . y f Ž ␩ .

,

igZ

where ␩ i, iq1 is the configuration that results from the jump of a single particle from site i to site i q 1,

␩

i , iq1

¡␩ Ž i . y 1, ¢␩ Ž j . ,

Ž j . s~␩ Ž i q 1 . q 1,

j s 1, j s i q 1, j / i , i q 1.

We shall construct the process through a graphical representation, so the generator is of interest only as a precise statement of the dynamics. The process is denoted by ␩ Ž t . s Ž␩ Ž i, t .: i g Z. where t G 0 is the time variable. We are interested in the scaling behavior of the particle density of the process. By suitably scaling space and time we expect to find a deterministic evolution around which the actual particle density fluctuates. Think of an asymmetric random walk. Its position has a nonrandom limit if space shrinks by a factor n, time speeds up by the same factor n and then n goes to infinity. This suggests the scaling for nontrivial behavior in the totally asymmetric K-exclusion. As we shall prove, under this scaling the particle density converges to a nonrandom density function uŽ x, t . that is determined by the initial density, by a conservation law, and by the entropy criterion for picking the relevant solution from the many possible weak solutions. To make this precise, suppose we have a sequence of K-exclusion processes ␩nŽ⭈., n s 1, 2, 3, . . . . The empirical particle density of the process is the random Radon measure on R defined by

Ž 2.2.

␲n Ž t . s

1 n

Ý ␩n Ž i , t . ␦i r n , igZ

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where ␦ x is a unit mass at the point x g R. To clarify, the integral of an element ␾ g C0 ŽR. Žthe space of continuous, compactly supported functions on R. against the measure ␲nŽ t . is given by 1

␲n Ž t , ␾ . s

n

Ý ␩n Ž i , t . ␾ Ž irn . . igZ

Radon measures are topologized weakly by such functions: ␮ n ª ␮ in the space of Radon measures on R if ␮ nŽ ␾ . ª ␮ Ž ␾ . for all ␾ g C0 ŽR.. Notice that ␲n already incorporates the space scaling that shrinks the lattice distance to 1rn. In the theorem below, this space scaling is complemented by the time scaling that requires us to look at ␲nŽ nt . instead of ␲nŽ t .. The final ingredient is the assumption that the initial distribution of the process has a well-defined macroscopic density. Let u 0 be a measurable function on R such that 0 F u0 Ž x . F K .

Ž 2.3.

Write Pn for the probability measure on the probability space of the process ␩nŽ⭈.. For all ␾ g C0 Ž R . and ␧ ) 0,

Ž 2.4.

lim Pn ␲n Ž 0, ␾ . y

nª⬁

ž

HR␾ Ž x . u

0

Ž x . dx G ␧ s 0.

/

This assumption says that the random measure ␲n satisfies a weak law of large numbers at time t s 0, and the limit is a deterministic measure u 0 Ž x . dx. The theorem is that such a law of large numbers continues to hold at later times. In addition, we prove a strong law under this assumption:

Ž 2.5.

The random initial configurations Ž␩nŽ i, 0.: i g Z., n g N, are all defined on a common probability space, and for all ␾ gC 0 ŽR., lim n ª ⬁ ␲nŽ0, ␾ . s HR ␾ Ž x . u 0 Ž x . dx almost surely.

We need a brief discussion about entropy solutions before stating the theorem. Let g K Ž x . be a nonincreasing, nonnegative convex function on R that satisfies

Ž 2.6.

g K Ž x . s 0 for x G 1 and

g K Ž x . s yKx for x F y1.

Given u 0 as above, pick a function U0 that satisfies

Ž 2.7.

U0 Ž b . y U0 Ž a . s

HŽ a , b x u

0

Ž x . dx for all a - b.

Then for x g R and t ) 0 define

Ž 2.8.

U Ž x , t . s sup U0 Ž y . y t g K ygR

½

ž

xyy t

/5

.

The supremum is in fact attained at some y g w x y t, x q t x . For each fixed t ) 0, UŽ⭈, t . is nondecreasing and Lipschitz continuous with Lipschitz con-


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stant K. All this follows from Ž2.3., 2.6. and Ž2.7., because by convexity g K has slope in w yK, 0x . Let uŽ x , t . s

Ž 2.9.

⭸ ⭸x

UŽ x , t . .

Define the flux function f K as the Legendre conjugate of Žthe negative of. g K :

Ž 2.10.

f K Ž ␳ . s inf ␳ x q g K Ž x . 4 , x

0 F ␳ F K.

Then UŽ x, t . is the unique viscosity solution of the Hamilton᎐Jacobi equation Ut q f K Ž Ux . s 0,

Ž 2.11.

U Ž x , 0 . s U0 Ž x . ,

while uŽ x, t . is the unique entropy solution of the scalar conservation law u t q f K Ž u . x s 0,

Ž 2.12.

u Ž x , 0. s u 0 Ž x . .

Equation Ž2.8. is well-known in the literature on partial differential equations. The names of Hopf, Lax and Oleinik are attached to it in various combinations by different authors. Here are some of the relevant references: Bardi and Evans Ž1984., Evans Ž1984., Hopf Ž1965., Lax Ž1957., Lions Ž1982., Ž1987.. and Lions, Souganidis and Vasquez ´ In our proof we use only formula Ž2.8. and nothing else about the partial differential equations. In fact, the centerpiece of our proof is a microscopic version of Ž2.8. wŽ 4.9. in Section 4x that emerges from the graphical construction of the K-exclusion. THEOREM 1. For each positive integer K, there exists a nonnegative convex function g K that satisfies Ž2.6., and such that this holds: Let uŽ x, t . be defined by Ž2.7. ᎐ Ž2.9.. Under assumption Ž2.4., for each t ) 0 the random measure ␲nŽ nt . converges in probability to uŽ x, t . dx. Precisely, for all ␾ g C0 ŽR. and ␧ ) 0,

Ž 2.13.

lim Pn ␲n Ž nt , ␾ . y

nª⬁

ž

HR␾ Ž x . u Ž x , t . dx

G ␧ s 0.

/

Under assumption Ž2.5. we can construct the processes ␩nŽ⭈. on a common probability space so that the strong law holds: ␲nŽ nt . ª uŽ x, t . dx almost surely as n ª ⬁, for all t ) 0. Properties of f K . Only in the case K s 1, the totally asymmetric simple exclusion process ŽTASEP., has f K been computed: f 1Ž ␳ . s ␳ Ž1 y ␳ .. This was first done by Rost Ž1981., for a special class of initial profiles. Presently we can prove that, if the initial distribution of the process is spatially ergodic with expectation Ew ␩ Ž i, 0.x s ␳ , then

Ž 2.14.

f K Ž ␳ . s lim

tª⬁

1

t

H P Ž ␩ Ž i , s . G 1, ␩ Ž i q 1, s . F K y 1. ds. t 0

ŽSee Section 9.. The scenario one expects is that for each density ␳ g w 0, K x there is a unique spatially ergodic equilibrium measure ␯␳ with expectation


367

␯␳ w ␩ Ž0.x s ␳ . From this and Ž2.14., it would follow that

Ž 2.15.

f K Ž ␳ . s ␯␳ Ž ␩ Ž i . G 1, ␩ Ž i q 1 . F K y 1 . .

Partial results on the existence and uniqueness of equilibria appear in the unpublished manuscript of Ekhaus and Gray Ž1994.. Here are some general properties of f K . Convexity of g K and definition Ž2.10. imply that f K is concave. On w 0, Kr2x f K is nondecreasing and on w Kr2, K x nonincreasing. We have symmetry fK Ž ␳ . s fK Ž K y ␳ .

Ž 2.16.

for ␳ g w 0, K x

and monotonic dependence on K : f K Ž ␳ . F f Kq1 Ž ␳ .

Ž 2.17.

for ␳ g w 0, K x .

Monotonicity, symmetry and the explicitly known f 1 give a lower bound,

Ž 2.18.

¡ ¢

␳ Ž1 y ␳ . , f K Ž ␳ . G 1r4, Ž K y ␳ . Ž1 y Ž K y ␳ . . ,

~

0 F ␳ F 1r2, 1r2 F ␳ F K y 1r2, K y 1r2 F ␳ F K .

An upper bound is obtained by comparison with the zero-range process,

Ž 2.19.

fK Ž ␳ . F

½

␳r Ž 1 q ␳ . , Ž K y ␳ . rŽ 1 q K y ␳ . ,

0 F ␳ F Kr2, Kr2 F ␳ F K .

The function g K is more fundamental to our approach than the flux f K , which is derived from g K by Ž2.10. after the proof is complete. In Section 5 we define g K as the asymptotic shape of a growth model that is closely associated with the K-exclusion process. Properties of both g K and f K are derived in Section 9. The marching soldiers model. As a corollary of our proof, we also get a hydrodynamic limit for a totally asymmetric marching soldiers model. The rules for this process are these: For each integer i g Z there is a soldier ␴ Ž i . whose location at time t is ␴ Ž i, t . g Z. Neighboring soldiers are not allowed to be too far apart, so there are constants L1 , L2 g Zq such that

Ž 2.20.

y L1 F ␴ Ž i q 1, t . y ␴ Ž i , t . F L2

for all i g Z and t G 0.

Subject to condition Ž2.20., the soldiers march forward at exponential rate 1, independently of each other. We state and prove a theorem that corresponds to the strong law part of Theorem 1. Suppose V0 is a function on R that satisfies

Ž 2.21.

y L 1 Ž x y y . F V 0 Ž x . y V0 Ž y . F L 2 Ž x y y .

for all y - x.

We make this assumption:

Ž 2.22.

A sequence of random initial configurations Ž ␴nŽ i, 0.: i g Z., n g N, is defined on a probability space, and for all y g R, lim n ª⬁Ž1rn. ␴nŽw ny x , 0. s V0 Ž y . almost surely.


368

THEOREM 2. Under assumption Ž2.22. we can construct the processes ␴nŽ⭈. on a common probability space so that the strong law continues to hold: lim nª⬁

1 n

␴n Ž w nx x , nt . s V Ž x , t .

a.s.

for all t ) 0 and x g R. The limit V Ž x, t . satisfies

Ž 2.23.

V Ž x , t . s inf V0 Ž y . y L1 Ž x y y . q t g L 1qL 2Ž Ž y y x . rt . 4 . ygR

The function g L 1qL 2 is the same as the one appearing in Theorem 1. By the remarks preceding Theorem 1, Theorem 2 implies that the motion of the soldiers’ front is governed by the differential equation ⭸ Vr⭸ t s f L 1qL 2Ž L1 q ⭸ Vr⭸ x ., where f L 1qL 2 is the function defined by Ž2.10.. If we consider just a single process w ␴nŽ⭈. s ␴ Ž⭈. for all n x , then Ž2.22. implies that V0 Ž y . s ␳ y for some yL1 F ␳ F L 2 . Equation Ž2.23. gives V Ž x, t . s ␳ x q tf L 1qL 2Ž L1 q ␳ .. We see that from an initial slope ␳ , each soldier has the same well-defined asymptotic speed,

Ž 2.24.

lim tª⬁

1 t

␴ Ž i , t . s V Ž 0, 1 . s f L 1qL 2Ž L1 q ␳ .

a.s.

for any fixed i. By Ž2.18., the asymptotic speed is nonzero if yL1 - ␳ - L2 . The MrMr1rK queueing picture. There is a natural queueing interpretation for K-exclusion. The sites represent a sequence of servers, and the particles are customers moving through the system. Servers serve at rate 1, and the queues at the servers have capacity K and FIFO discipline. The ordering of customers is preserved by this convention: The queue of ␩ Ž i . customers at server i is represented by an ordered stack, with the oldest customer at the bottom and currently in service and with the most recently arrived customer at the top. When a jump from site i occurs, the bottom customer of queue i leaves Žhis service was completed at server i ., and he becomes the top customer of queue i q 1. In Section 4 we introduce another particle system where the customers are fixed in space and the servers jump past the customers. To keep the two processes separate we shall call the K-exclusion ␩ Ž⭈. the customer process. 3. Disordered K-exclusion. The disordered process operates as the one described in Section 2, except for this difference: the rate of jumping from site i is now a number ␣ Ž i ., instead of a uniform rate 1 for all sites. Let ␣ s Ž ␣ Ž i .: i g Z. denote the sequence of jump rates. Once ␣ is picked, the generator Ž2.1. becomes

Ž 3.1.

L␣ f Ž ␩ . s

Ý ␣ Ž i . I␩ Ž i.G 1 , ␩ Ž iq1.F Ky14 f Ž␩ i , iq1 . y f Ž ␩ .

.

igZ

We assume that a0 F ␣ Ž i . F 1 for all i g Z, where a0 ) 0 is a fixed constant. Thus ␣ is an element of the space A s w a0 , 1x Z . We want random

369


jump rates, so we assume that we have a probability measure Q on A under which the ␣ Ž i .’s are i.i.d. random variables. The marginal distribution of ␣ Ž0. on w a0 , 1x is arbitrary. The idea of quenched disorder is that ␣ , though random, is picked once and fixed for the duration of the dynamics. So, in the setting of a sequence of processes ␩nŽ⭈. described in Section 2, we assume that a choice of ␣ g A is made at the outset, and then each process ␩nŽ⭈. operates under these same rates, so that jumps from site i always occur at rate ␣ Ž i .. Write Pn␣ for the probability measure of the nth process ␩nŽ⭈. with rates ␣ . We only prove a weak law here, and the assumption is the same as for Theorem 1. For all ␾ g C0 Ž R . and ␧ ) 0,

Ž 3.2.

lim Pn␣ ␲n Ž 0, ␾ . y

ž

nª⬁

HR␾ Ž x . u

0

Ž x . dx G ␧ s 0.

/

Let a1 s Ž E Q w ␣ Ž0.y1 x.y1 g w a 0 , 1x . Property Ž2.6. of the basic setting now becomes

Ž 3.3.

g K Ž y . s 0 for y G a1

and

g K Ž y . s yKy for y F ya1 .

THEOREM 3. Fix a positive integer K and an i.i.d. distribution Q for ␣ s Ž ␣ Ž i ... Then there exists a nonnegative convex function g K that depends on Q and has property Ž3.3., and a subset A0 : A such that QŽ A0 . s 1 and such that the following statement holds for g K and all ␣ g A0 : if assumption Ž3.2. is satisfied and uŽ x, t . is defined by Ž2.7. ᎐ Ž2.9., then for each t ) 0, ␾ g C0 ŽR., and ␧ ) 0:

Ž 3.4.

lim Pn␣ ␲n Ž nt , ␾ . y

nª⬁

ž


G ␧ s 0.

/

We emphasize that the good set A0 of disorder variables ␣ does not depend on the initial distributions of the processes ␩nŽ⭈.. The way this set is constructed can be seen from Proposition 5.2 below. Notice also this: Even though assumption Ž3.2. is stated in terms of the measures Pn␣ , the disorder variable ␣ really has nothing to do with the initial distribution of the process. The rates ␣ are fixed, and then the process is constructed on the product probability space of the initial distribution and of the Poisson processes Di 4 of the graphical construction Žsee Section 4.. It is of course possible to look at the process simultaneously under many or all ␣ ’s and to pick initial distributions that depend on ␣ . Then Theorem 3 says that Ž3.4. holds for each ␣ in A0 for which assumption Ž3.2. is valid. The assumption that Ž ␣ Ž i .. are i.i.d. as opposed to merely ergodic is used only in the proof of Lemma 5.4, to obtain a probability estimate good enough for the Borel᎐Cantelli lemma. Consequently, any mixing assumption on Ž ␣ Ž i .. strong enough to give a summable bound in Lemma 5.4 is good enough for our proof of Theorem 3. The existence of g K , and thereby the existence of f K through Ž2.10., requires only ergodicity for Ž ␣ Ž i ... ŽSee Proposition 5.1.. Ergodicity of Ž ␣ Ž i .. is sufficient for K-monotonicity Ž2.17. of f K . We prove


370

symmetry Ž2.16. under reversibility of Ž ␣ Ž i .. ŽLemma 9.1., which we have in the i.i.d case. A recent result of Goldstein and Speer Ž1998. suggests that symmetry may hold much more generally. If the initial distribution of the process is spatially ergodic with expectation Ew ␩ Ž i, 0.x s ␳ , then Ž2.14. takes the form 1 t ␣ P Ž ␩ Ž 0, s . G 1, ␩ Ž 1, s . F K y 1 . ds, Ž 3.5. f K Ž ␳ . s ␣ Ž 0 . lim tª⬁ t 0 for any ␣ g A0 . The reader may find it somewhat surprising that the righthand side of Ž3.5. should converge to something independent of ␣ . We can check the reasonableness of this contention by comparing with a sitedisordered zero-range process for which explicit calculations are possible. If we take K s ⬁ in the generator Ž3.1., we get a zero-range process with product-form equilibria ␯␳␣ . Under ␯␳␣ the variables Ž␩ Ž i .: i g Z. are mutually independent with geometric marginals

H

k Ž 3.6. ␯␳␣ Ž ␩ Ž i . s k . s Ž 1 y vr␣ Ž i . . Ž vr␣ Ž i . . ,

k s 0, 1, 2, 3, . . . .

Here v g Ž0, a0 . is the average jump rate, common to all sites, and determined by the density ␳ through the equation ␳ s E Q w vrŽ ␣ Ž0. y v .x . w See Benjamini, Ferrari and Landim Ž1996. or Seppalainen and Krug Ž1998..x If ¨¨ the zero-range process is started with ␯␳␣ , then the right-hand side of Ž3.5. becomes ␣ Ž0.␯␳␣ Ž␩ Ž0. G 1. s v, as expected. In case the reader wonders about the necessity to assume a0 ) 0: if we allowed rates arbitrarily close to zero, the particle profile would not move on the hydrodynamic scale. See the last paragraph of Section 6. 4. The coupling. We begin by constructing the disordered K-exclusion process. Pick and fix the rates ␣ s Ž ␣ Ž i ... The reader only interested in the basic case without disorder should set ␣ Ž i . ' 1 throughout this section. As hinted in the remark about the queueing interpretation in Section 2, we do the construction w and prove the theoremsx through the corresponding server process z Ž⭈.. This process z Ž t . s Ž z Ž i, t .: i g Z. satisfies the following rules: Ž1. Its state is an ordered, labeled particle configuration z s Ž z Ž i .: i g Z. g Z Z with z Ž i . F z Ž i q 1. F z Ž i . q K for all i g Z. Ž2. The dynamics is determined by a collection Di 4 of mutually independent Poisson processes on w 0, ⬁., where Di has rate ␣ Ž i .. At the epochs of Di , z Ž i . attempts to jump one step to the left, and the jump is executed unless it threatens to violate the inequalities z Ž i . G z Ž i y 1. and z Ž i . G z Ž i q 1. y K. It is clear that if we construct a process z Ž⭈. with these two properties, then the disordered K-exclusion of Section 3 can be constructed by defining ␩ Ž i , t . s z Ž i , t . y z Ž i y 1, t . . Ž 4.1. In other words, particle Žserver. z Ž i . jumps precisely when a K-exclusion particle Žcustomer. jumps from site i to i q 1. All our remaining work will be in terms of the server process z Ž⭈., and the results for the customer process follow from Ž4.1..


Construction of Assume given an above. The bound tions are valid for

371

the server process z Ž⭈.. This goes by standard arguments. initial configuration Ž z Ž i, 0.: i g Z. that satisfies rule Ž1. ␣ Ž i . F 1 on the rates assures that the following assumpalmost every realization of the Poisson point processes:

Ž4.2a. The Di 4 are such that there are no simultaneous jump attempts. Ž4.2b. Each Di has only finitely many epochs in any bounded time interval. Given any t 1 ) 0, there are arbitrarily faraway indices i 0 < 0 < i 1 Ž4.2c. such that neither Di nor Di has any epochs in the time interval 0 1 w 0, t 1 x . Then z Ž i 0 . and z Ž i 1 . do not attempt to jump during w 0, t 1 x , and the evolution of the servers z Ž i 0 q 1., . . . , z Ž i 1 y 1.4 is isolated from the rest of the process up to time t 1. There are only finitely many jump attempts in D i 0 F i F i 1 Di l w 0, t 1 x , so they can be ordered, and then the evolution of z Ž i, t ., for i 0 F i F i 1 and 0 F t F t 1 , can be computed by considering the jump attempts in their temporal order and applying rule Ž2.. This completes the construction of z Ž⭈.. With ␣ s Ž ␣ Ž i .. and the initial Ž z Ž i, 0.. fixed, the process is defined on the probability space of the Di 4 . This construction can be used in a standard way to produce couplings that preserve order. Proof of the following lemma is left to the reader. LEMMA 4.1. Pick a realization Di 4 of the Poisson processes that satisfies properties Ž4.2a. ᎐ Ž4.2c.. Suppose w Ž t . s Ž w Ž i, t .: i g Z. and y Ž t . s Ž y Ž i, t .: i g Z. are server processes constructed exactly as described above, from initial configurations Ž w Ž i, 0.: i g Z. and Ž y Ž i, 0.: i g Z., and so that for each i, server particles w Ž i . and y Ž i . read their jump attempts from Di . Then if initially w Ž i, 0. F y Ž i, 0. for all i g Z, the same ordering w Ž i, t . F y Ž i, t . holds for all later times t G 0 and for all i g Z. Write w l Ž⭈. for the server process with this special initial configuration:

Ž 4.3.

w l Ž j, 0 . s

½

l, l q Kj,

j G 0, j - 0.

These processes occupy a special role in our development, hence the distinct notation. From the full collection w l Ž⭈.: l g Z4 , we actually construct only the subcollection w zŽ i, 0. Ž⭈.: i g Z4 for a given initial configuration Ž z Ž i, 0.: i g Z. of the z Ž⭈.-process. Once the rates ␣ s Ž ␣ Ž i .. and a realization Di 4 of the Poisson event times have been chosen, we construct the processes w zŽ i, 0. Ž⭈. so that particle w zŽ i, 0. Ž j . reads its jump attempts from the Poisson process Diqj . In particular,

Ž 4.4.

the jump rate of particle w zŽ i , 0. Ž j . is ␣ Ž i q j . .

Now z Ž⭈. and w z Ž i, 0. Ž⭈.4 are coupled, that is, constructed on the same probability space. We have the following crucial lemma.

372


LEMMA 4.2. Assume that Di 4 satisfies assumptions Ž4.2a. ᎐ Ž4.2c.. Then the following equality holds for all k g Z and t G 0:

Ž 4.5.

z Ž k , t . s sup w zŽ i , 0. Ž k y i , t . . igZ

PROOF. Choosing i 0 and i 1 for t 1 as in Ž4.2c., it suffices to prove inductively that Ž4.5. holds after each epoch of D i 0 - k - i 1 Dk l w 0, t 1 x , for i 0 - k - i 1. Notice that all the particles involved in Ž4.5. for a particular k read their jump commands from Dk , so the validity of Ž4.5. for k cannot change at epochs of Dj for j / k. Equation Ž4.5. certainly holds at time t s 0; the supremum is realized by i s k, by any i ) k such that z Ž i, 0. s z Ž k, 0. q Ž i y k . K and by any i - k such that z Ž i, 0. s z Ž k, 0.. Also, Ž4.5. holds for k s i 0 and k s i 1 for all t g w 0, t 1 x because none of the particles involved even attempts to jump during this time interval. Now suppose ␶ is an epoch of the Poisson process Dk for some i 0 - k - i 1 , and assume by induction that Ž4.5. holds for all i 0 F k F i 1 and t g w 0, ␶ .. Three cases need to be considered. CASE 1 w z Ž k . jumps at time ␶ x . We must show that if w zŽ i, 0. Ž k y i, ␶ y . s z Ž k, ␶ y ., then w zŽ i, 0. Ž k y i . also jumps at time ␶ . Since z Ž k . can execute its jump, we must have z Ž k y 1, ␶ y . F z Ž k, ␶ y . y 1 and z Ž k q 1, ␶ y . F z Ž k, ␶ y . q K y 1. By the induction assumption, w zŽ i , 0. Ž k y i y 1, ␶ y . F z Ž k y 1, ␶ y . F z Ž k , ␶ y . y 1 s w zŽ i , 0. Ž k y i , ␶ y . y 1 and w zŽ i , 0. Ž k y i q 1, ␶ y . F z Ž k q 1, ␶ y . F z Ž k , ␶ y . q K y 1 s w zŽ i , 0. Ž k y i , ␶ y . q K y 1, so that w zŽ i, 0. Ž k y i . can also jump at time ␶ . Since this argument applies to any i such that w zŽ i, 0. Ž k y i, ␶ y . s z Ž k, ␶ y ., Ž4.5. continues to hold for k at time ␶ . CASE 2 w z Ž k . does not jump at time ␶ because z Ž k y 1, ␶ y . s z Ž k, ␶ y .x . By induction there exists an i such that z Ž k y 1, ␶ y . s w zŽ i, 0. Ž k y i y 1, ␶ y ., and consequently, again by induction, z Ž k , ␶ y . s z Ž k y 1, ␶ y . s w zŽ i , 0. Ž k y i y 1, ␶ y . F w zŽ i , 0. Ž k y i , ␶ y . F z Ž k , ␶ y . , from which we conclude that w zŽ i, 0. Ž k y i, ␶ y . s z Ž k, ␶ y . and that w zŽ i, 0. Ž k y i . cannot jump at time ␶ because it is blocked by w zŽ i, 0. Ž k y i y 1.. Thus z Ž k, ␶ . s w zŽ i, 0. Ž k y i, ␶ ., and Ž4.5. continues to hold for k at time ␶ .


373

CASE 3 w z Ž k . does not jump at time ␶ because z Ž k q 1, ␶ y . s z Ž k, ␶ y . q K x . This case is in principle like the previous one, and we leave the details to the reader. I An application of Lemma 4.1 to w Ž k, t . s w z Ž j, 0. Ž k y j, t . and y Ž k, t . s w k y i, t . q z Ž j, 0. y z Ž i, 0. gives the next lemma. zŽ i, 0. Ž

LEMMA 4.3. w

For i - j, all k, and all t G 0,

zŽ i , 0.

Ž k y i , t . G w zŽ j, 0. Ž k y j, t . y z Ž j, 0 . y z Ž i , 0 . .

The evolution of the processes w zŽ i, 0. Ž⭈. comes from two sources: the initial configuration Ž z Ž i, 0.: i g Z. that determines the initial configurations Ž w zŽ i, 0. Ž j, 0.: j g Z., and the increments w zŽ i, 0. Ž j, t . y w zŽ i, 0. Ž j, 0. that come from the Poisson processes. To explicitly separate these, define the nonnegative, increasing processes ␰ zŽ i, 0. Ž⭈. by

␰ zŽ i , 0. Ž j, t . s z Ž i , 0 . y w zŽ i , 0. Ž j, t . , j g Z, t G 0. zŽ i, 0. We regard ␰ as the position of an interface that moves to the right on the plane. At time t, ␰ zŽ i, 0. Ž j, t . is the distance at level j from the y-axis to the interface. Once the servers w zŽ i, 0. Ž j . have been assigned rates as in Ž4.4., the dynamics of ␰ zŽ i, 0. is determined by the rules of the server process w zŽ i, 0.. Initially

Ž 4.6.

␰ zŽ i , 0. Ž j, 0 . s 0 for j G 0 and ␰ zŽ i , 0. Ž j, 0 . s yKj for j - 0. Then ␰ zŽ i, 0. Ž j . jumps one step to the right at rate ␣ Ž i q j ., under the restrictions

Ž 4.7.

␰ zŽ i , 0. Ž j, t . F ␰ zŽ i , 0. Ž j y 1, t . F ␰ zŽ i , 0. Ž j, t . q K . By Ž4.6. the process ␰ zŽ i, 0. Ž⭈. registers only increments of the w zŽ i, 0. Ž⭈.-process and not absolute location. In particular, the distribution of the process ␰ zŽ i, 0. Ž⭈. does not depend on the actual location z Ž i, 0. in the initial z-configuration, and this distribution depends on the index i only by a translation of the rates ␣ . Consequently, in the basic case without disorder w ␣ Ž j . ' 1x , the processes ␰ zŽ i, 0. Ž⭈.: i g Z4 are identically distributed. With these interface processes, we rewrite Ž4.5. as

Ž 4.8.

z Ž k , t . s sup z Ž i , 0 . y ␰ zŽ i , 0. Ž k y i , t . 4 .

Ž 4.9.

igZ

This equation is the basis for all our results. In the next section we develop an alternative definition for the process ␰ to prove its scaling limit. As the final point of this section we state one lemma about Ž4.9. for future use. LEMMA 4.4.

␰

zŽ j1 , 0.

Suppose there are j1 - k - j2 such that

Ž k y j1 , t . s 0 and ␰ zŽ j 2 , 0. Ž k y j2 , t . s yK Ž k y j2 . .

Then z Ž k , t . s max

j1FiFj 2

z Ž i , 0. y ␰ zŽ i , 0. Ž k y i , t . 4 .


374

PROOF. For i - j1 , z Ž i , 0 . y ␰ zŽ i , 0. Ž k y i , t . F z Ž i , 0 . F z Ž j1 , 0 . s z Ž j1 , 0 . y ␰ zŽ j1 , 0. Ž k y j1 , t . , and consequently z Ž k , t . s sup z Ž i , 0 . y ␰ zŽ i , 0. Ž k y i , t . 4 . i: iGj1

We leave the other half to the reader. I 5. The growth model. In this section we study the planar growth model encountered at the end of the previous section. We develop a ‘‘last-passage’’ formulation for it Žas opposed to ‘‘first-passage’’ percolation. to apply the subadditive ergodic theorem. After Proposition 5.1 we define the function g K that appears in Theorems 1᎐3. The reader who wishes to consider only the basic case of Section 2, without disorder, should set ␣ Ž j . ' 1 throughout this section. For each K there is a notion of a K-admissible path. A K-admissible path on the lattice Z 2 is a finite sequence r s ␬ 1 , ␬ 2 , . . . , ␬ m 4 of sites or points of Z 2 that satisfy this constraint: There are three admissible kinds of steps in the path. For each i s 1, . . . , m y 1, ␬ iq1 y ␬ i s Ž 1, 0 . , Ž 0, 1 . , or Ž K , y1 . . Ž 5.1. Geometrically speaking, a K-admissible path can take steps up, steps right, and chess knight-type steps K lattice increments to the right and one down. ŽIn the case K s 2 this is exactly a knight’s move in chess. For K s 1 it is actually a bishop’s move diagonally southeast, while for K G 3 it is a ‘‘stretched’’ knight’s move.. The empty path with m s 0 is also deemed K-admissible. Let LK denote the set of integer sites that can be reached from the point Ž1, 0. along K-admissible paths. Quite obviously LK s Ž k , l . g N = Z: l G y Ž k y 1 . rK 4 . This set is a wedge bounded by the positive y-axis and a lattice equivalent of the line with slope y1rK. All K-admissible paths started inside LK remain inside LK . Next we introduce the passage times of sites and paths. Let ␭ s Ž ␭Ž i, j .: Ž i, j . g Z 2 . be an assignment of nonnegative numbers to the sites of Z 2 . To the points of the lattice y-axis, attach the quenched rate variables ␣ s Ž ␣ Ž j .: j g Z.. The passage time of site Ž i, j . is then ␶ Ž i, j . s ␣ Ž j .y1␭Ž i, j .. The 2 configuration ␭ is an element of the space S s w 0, ⬁. Z , while the sequence ␣ Z is an element of the space A s w a0 , 1x as defined in Section 3. Elements of the product space ⌺ s S = A are indexed by Z 2 in the obvious way: Ž ␭ , ␣ .Ž i, j . s Ž ␭Ž i, j ., ␣ Ž j ... Define translation operators ⌰Ž k, l ., Ž k, l . g Z 2 , on ⌺ by ⌰ Ž k , l . Ž ␭ , ␣ . Ž i , j . s Ž ␭Ž i q k , j q l . , ␣ Ž j q l . . . Ž 5.3.

Ž 5.2.


375

Let ␮ be the probability measure on S under which the Ž ␭Ž i, j .. are i.i.d. ExpŽ1.-distributed, and let Q be the i.i.d. measure on A as denoted in Section 3. The probability measure we put on ⌺ is ⺠ s ␮ m Q, the product measure under which ␭ and ␣ are mutually independent. Expectation under ⺠ is denoted by ⺕. When ␣ is fixed, ⺠ ␣ s ␮ m ␦␣ is the conditional distribution on ⌺. Under ⺠ ␣ , the passage time ␶ Ž i, j . is ExpŽ ␣ Ž j ..-distributed. This is quenched disorder, where the ␣ Ž j .’s are fixed but the ␭Ž i, j .’s remain random. However, at this stage we regard both ␭ and ␣ random. The passage time of a path r s ␬ 1 , ␬ 2 , . . . , ␬ m 4 is simply the sum of the passage times of the sites along the path m

Ž 5.4.

TŽ r. s

Ý ␶ Ž ␬i . . is1

For any two integer points Ž u1 , v 1 . and Ž u 2 , v 2 . on the plane such that Ž u 2 , v 2 . g Ž u1 , v 1 . q LK , let RK ŽŽ u1 , v 1 ., Ž u 2 , v 2 .. denote the collection of Kadmissible paths r s ␬ 1 , ␬ 2 , . . . , ␬ m 4 such that ␬ 1 s Ž u1 q 1, v1 . and ␬ m s Ž u 2 , v 2 ., and m of course may vary. Our convention is ␬ 1 s Ž u1 q 1, v 1 . instead of ␬ 1 s Ž u1 , v 1 . so that elements of RK ŽŽ u1 , v 1 ., Ž u 2 , v 2 .. and RK ŽŽ u 2 , v 2 ., Ž u 3 , v 3 .. can be joined to form an element of RK ŽŽ u1 , v 1 ., Ž u 3 , v 3 .. without repeating the point Ž u 2 , v 2 . The basic case is Ž u1 , v 1 . s Ž0, 0., in which case paths start at Ž1, 0.. The passage time from Ž u1 , v1 . to Ž u 2 , v 2 . is defined by

Ž 5.5. TK Ž Ž u1 , v1 . , Ž u 2 , v 2 . . s max T Ž r . : r g RK Ž Ž u1 , v1 . , Ž u 2 , v 2 . . 4 . Abbreviate TK Ž u, v . s TK ŽŽ0, 0., Ž u, v ... The growth model is a randomly growing subset of LK , defined by proclaiming that point Ž i, j . joins the growing cluster at time TK Ž i, j .. So, if B Ž t . denotes the cluster at time t, the rule is

Ž 5.6.

B Ž t . s Ž i , j . g LK : TK Ž i , j . F t 4 .

A random process ␰ marks the location of the interface of B Ž t ., in the sense that

Ž 5.7.

B Ž t . s Ž i , j . g LK : i F ␰ Ž j, t . 4 .

So the interface moves to the right, in the increasing x-direction. For j g Z and t G 0, ␰ Ž j, t . denotes the distance from point j on the y-axis to the interface on the right-hand side of this axis. In terms of the passage times,

Ž 5.8.

␰ Ž j, t . s min i: Ž i q 1, j . g LK , TK Ž i q 1, j . ) t 4 .

Initially ␰ Ž j, 0. s 0 for j G 0 and ␰ Ž j, 0. s yKj for j - 0. At this point the reader should observe Lemma 5.1. LEMMA 5.1. Fix K and the rates ␣ . The distribution of the process ␰ Ž⭈. under ⺠ ␣ is equal to the distribution of the process ␰ zŽ0, 0. Ž⭈. defined by Ž4.6., for any random or deterministic initial configuration Ž z Ž i, 0.: i g Z..


376

Indeed, the whole point of the definition of K-admissible paths is that the evolution of the surface ␰ Ž⭈. replicates the evolution of ␰ zŽ0, 0. Ž⭈. that comes from the server process. We prove Lemma 5.1 in a more general setting in Section 10 w see the proof following Ž10.36.x , so we will not waste space on it here. But this should convince the reader: according to Ž5.8. and the definition of K-admissible paths, ␰ Ž j, t . increases from i to i q 1 at rate ␣ Ž j . provided

␰ Ž j, t . s i , ␰ Ž j y 1, t . G i q 1 and

␰ Ž j q 1, t . G i y K q 1. On the other hand, by Ž4.6. and the definition of the server dynamics, ␰ zŽ0, 0. Ž j, t . increases from i to i q 1 precisely when w zŽ0, 0. Ž j, t . jumps from z Ž0, 0. y i to z Ž0, 0. y i y 1, and this happens at rate ␣ Ž j . provided w zŽ0 , 0. Ž j, t . s z Ž 0, 0 . y i , w zŽ0 , 0. Ž j y 1, t . F z Ž 0, 0 . y i y 1 and w zŽ0 , 0. Ž j q 1, t . F z Ž 0, 0 . q K y i y 1, or, equivalently, provided

␰ zŽ0 , 0. Ž j, t . s i , ␰ zŽ0 , 0. Ž j y 1, t . G i q 1 and

␰ zŽ0 , 0. Ž j q 1, t . G i y K q 1. Thus both the initial condition and the infinitesimal rates for ␰ Ž⭈. and ␰ zŽ0, 0. Ž⭈. are the same. Let UK s Ž x , y . g R 2 : x ) 0, y ) yxrK 4 be the continuum analogue of the lattice LK . PROPOSITION 5.1. There exists a finite, concave function ␥ K Ž x, y . defined on UK and a subset ⌺ 0 : ⌺ such that ⺠Ž ⌺ 0 . s 1, and 1 lim TK Ž w nx x , w ny x . s ␥ K Ž x , y . Ž 5.9. nª⬁ n holds for all Ž x, y . g UK and all Ž ␭ , ␣ . g ⌺ 0 . Furthermore, ␥ K can be extended to a continuous function on the closure UK . PROOF. Fix first a lattice point Ž k, l . g LK . The limit Ž5.9. for Ž x, y . s Ž k, l . will come from the subadditive ergodic theorem, so the first task is the moment bound ⺕ TK Ž nk , nl . F Cn Ž 5.10. for some constant C, for all n.


377

Begin by observing that

Ž 5.11.

TK Ž i , j . F T1 Ž i , j .

for any Ž i, j . g LK . This is because a K-admissible path can be turned into a 1-admissible path by replacing each Ž K, y1.-step by K y 1 Ž1, 0.-steps and one Ž1, y1.-step. This transformation cannot decrease the passage time of the path because no sites are removed. To bound T1Ž i, j . for Ž i, j . g L1 , define a bijection p: L1 ª N 2 by pŽ i, j . s Ž i, i q j .. A 1-admissible path from Ž1, 0. to Ž i, j . in L1 becomes under p a path in N 2 that connects Ž1, 1. to Ž i, i q j . and takes steps up, to the right and diagonally northeast. A northeast step can be replaced by a step up followed by a step right, so this image path is contained in a path from Ž1, 1. to Ž i, i q j . that takes steps only up and to the right. Define i.i.d. ExpŽ1. passage times v Ž i, j . for sites Ž i, j . g N 2 by v Ž i, j . s ␭Ž py1 Ž i, j ... Ž py1 is the inverse mapping.. For Ž i, j . g N 2 , let

Ž 5.12.

V Ž i , j . s max r⬘

Ý

v Ž s, t . ,

Ž s, t .gr ⬘

where the maximum is over up-right paths r ⬘ from Ž1, 1. to Ž i, j .. In other words, if r ⬘ s Ž s1 , t 1 ., . . . , Ž sm , t m .4 , then Ž s1 , t 1 . s Ž1, 1., Ž sm , t m . s Ž i, j . and for l s 1, . . . , m y 1, Ž slq1 , t lq1 . y Ž sl , t l . s Ž1, 0. or Ž0, 1.. The steps we took to construct the passage times V Ž i, j ., and the bound ␶ Ž i, j . s ␣ Ž j .y1␭Ž i, j . F Ž . ay1 0 ␭ i, j imply that

Ž 5.13.

T1 Ž i , j . F ay1 0 V Ž i, i q j. .

The advantage gained is that V Ž i, j . is easy to bound. We insert a lemma. LEMMA 5.2. We have the stochastic dominance V Ž i, j . F S 1r2 Ž i q j ., where S m. denotes a sum of m i.i.d. ExpŽ1r2. random variables. 1r2 Ž

PROOF. We appeal to a well-known MrMr1 queueing interpretation of V Ž i, j . w see Section 2 in Glynn and Whitt Ž1991.x . Imagine j MrMr1 servers in series, each with service rate 1, unlimited waiting space, and FIFO queueing discipline. At time 0, i customers are in queue at server 1, while servers 2, . . . , j have empty queues. Each customer moves through the entire series of j servers, joining the queue at server l q 1 as soon as service with server l is completed. Let v Ž k, l . denote the service time of customer k s 1, . . . , i at server l s 1, . . . , j. Then one can show inductively that V Ž i, j . defined by Ž5.12. is the time when customer i leaves server j. Next assume that, instead of queueing up at server 1 at time 0, customers 1, . . . , i arrive at server 1 in a PoissonŽ1r2. process, and that furthermore, at time 0 the queues at servers 1, . . . , j are not empty, but are i.i.d. GeomŽ1r2.distributed. The system starts in equilibrium and consequently stays in equilibrium. w See Kelly Ž1979. for more on this.x The waiting times an individual customer experiences at the successive queues are i.i.d. ExpŽ1r2.distributed. Thus the time for customer i to leave server j is S 1r2 Ž i q j .,


378

because S 1r2 Ž i . is the time when customer i arrives in queue 1 in the PoissonŽ1r2. arrival process, and S 1r2 Ž j . is the time it takes customer i to get through j queues. Compared to the original system, customer i is slowed down. This is made precise by a coupling argument. The upshot is that V Ž i, j . is stochastically dominated by S 1r2 Ž i q j .. I REMARK 5.1. The process V Ž i, j . has been studied by several authors. Glynn and Whitt Ž1991. derive a moment bound for V Ž i, j . with an elegant associativity argument. The limit

Ž 5.14.

lim nª⬁

1 n

V Ž w nx x , w ny x . s Ž 'x q y .

'

2

2 a.s. for Ž x , y . g Rq

was first obtained by Rost Ž1981.. We return to the proof of Proposition 5.1. Lemma 5.2 implies that

Ž 5.15.

⺕ V Ž i , j . F 2Ž i q j .

which, through Ž5.11. and Ž5.13., gives Ž5.10.. Now define X m, n s yTK ŽŽ mk, ml ., Ž nk, nl .. for 0 F m - n. We leave it to the reader to verify that Ž X m, n . satisfies the assumptions of a suitable subadditive ergodic theorem, such as that in Chapter 6 of Durrett Ž1991.. The conclusion is that for all Ž k, l . g LK , there is a finite number ␥ K Ž k, l . such that Ž5.9. holds ⺠-a.s. for Ž x, y . s Ž k, l .. Next we extend the limit and the definition of ␥ K , first to rational Ž x, y ., and then to all Ž x, y .. First check that we have homogeneity: ␥ K Ž mk, ml . s m␥ K Ž k, l . for m g N. This implies that for Ž x, y . g UK l Q 2 , we can unambiguously define

Ž 5.16.

␥K Ž x , y . s

1 m

␥ K Ž mx , my .

for any m g N that satisfies Ž mx, my . g LK . Keep such an m fixed, and for arbitrary n, let k be such that km F n - Ž k q 1. m. Then TK Ž kmx, kmy . F TK Ž w nx x , w ny x . F TK Ž Ž k q 1 . mx , Ž k q 1 . my . , and by letting n, k ª ⬁, we get the limit Ž5.9. a.s. for rational Ž x, y . g UK . Check that we have superadditivity

Ž 5.17.

␥ K Ž x 1 q x 2 , y1 q y 2 . G ␥ K Ž x 1 , y1 . q ␥ K Ž x 2 , y 2 . ,

homogeneity

Ž 5.18.

␥ K Ž sx 1 , sy1 . s s␥ K Ž x 1 , y 1 .

and, consequently, concavity

Ž 5.19.

␥ K Ž ␪ x 1 q Ž 1 y ␪ . x 2 , ␪ y1 q Ž 1 y ␪ . y 2 . G ␪␥ K Ž x 1 , y 1 . q Ž 1 y ␪ . ␥ K Ž x 2 , y 2 . ,

379


for rational Ž x 1 , y 1 ., Ž x 2 , y 2 . g UK , s ) 0 and ␪ g Ž0, 1.. Quite obviously also, if Ž x, y ., Ž x⬘, y⬘. g UK l Q 2 are such that Ž x⬘, y⬘. / Ž x, y . and Ž x⬘, y⬘. g Ž x, y . q UK , then

␥ K Ž x , y . - ␥ K Ž x⬘, y⬘ . .

Ž 5.20.

We need a continuity result which will be given in Lemma 5.3. LEMMA 5.3.

Ž 5.21.

For Ž x, y . g UK l Q 2 , lim ␥ K Ž x⬘, y . s ␥ K Ž x , y . .

Q2x ⬘ox

PROOF. If y s 0, the statement follows from homogeneity Ž5.18.. Suppose y ) 0 and x⬘ ) x. Then because Ž x, y . g Ž x, xyrx⬘. q UK , we have by Ž5.20. and Ž5.18. that

␥ K Ž x⬘, y . G ␥ K Ž x , y . G ␥ K Ž x , yxrx⬘ . s Ž xrx⬘ . ␥ K Ž x⬘, y . . The result follows by letting x⬘ o x. For y - 0, the result follows by the same principle. Take x⬙ s x⬘Ž y q xrK .rŽ y q x⬘rK . and y⬙ s yx⬙rx ⬘. Then Ž x, y . g Ž x⬙, y⬙ . q UK , so that

␥ K Ž x , y . G ␥ K Ž x⬙ , y⬙ . s Ž x⬙rx ⬘ . ␥ K Ž x⬘, y . , and again we may let x⬘ o x which also sends x⬙rx ⬘ ª 1. This proves Ž5.21.. I We return to the proof of Proposition 5.1. Define the set ⌺ 0 in the statement of the proposition as that subset of ⌺ on which the limit Ž5.9. holds for all rational Ž x, y . g UK . Extend ␥ K Ž x, y . to all Ž x, y . g UK by the formula

Ž 5.22. ␥ K Ž x , y . s inf ␥ K Ž x⬘, y⬘ . : Ž x⬘, y⬘ . g Ž x , y . q UK l Q 2 4 . On the right-hand side are the values ␥ K Ž x⬘, y⬘. for rational Ž x⬘, y⬘. already defined by Ž5.9.. By Ž5.20. and Ž5.21., for Ž x, y . g UK l Q 2 , Ž5.22. is a valid identity, so it is a sensible extension of ␥ K . Superadditivity Ž5.17. and homogeneity Ž5.18. can be proved again, this time for all Ž x, y . g UK . From this follows concavity. A finite, concave function is continuous on an open, convex set w Rockafellar Ž1970., Theorem 10.1x . Consequently, ␥ K Ž x, y . is continuous on UK . Continuity at boundary points Ž x, y . g UK _ UK requires separate arguments in the style of Lemma 5.3, and we leave them to the reader. The final point is the validity of the limit Ž5.9. for all Ž x, y . g UK and all Ž ␭ , ␣ . g ⌺ 0 . This follows from the continuity of ␥ K , by approximating a general lattice point Žw nx x , w ny x. by Žw nx⬘x , w ny⬘x. with rational Ž x⬘, y⬘.. The proof of Proposition 5.1 is complete. I Since ␥ K Ž x, y . is strictly increasing in x and y, the following equation defines uniquely a finite, convex, continuous, nonincreasing function g K on R: g K Ž y . s inf x ) 0: Ž x , y . g UK , ␥ K Ž x , y . G 1 4 ,

y g R.


380

By applying the strong law to the passage times ␶ Ž1, j ., j G 0, and ␶ ŽyKj, j ., j F y1, we get

Ž 5.23.

␥ K Ž 0, y . G yra1

and ␥ K Ž Ky, yy . G yra1 for y G 0.

This implies Ž2.6. and Ž3.3.. REMARK 5.2. In the basic case without disorder w ␣ Ž j . ' 1x , we can obtain the exact boundary values

␥ K Ž 0, y . s ␥ K Ž Ky, yy . s y for y G 0.

Ž 5.24.

This is proved in Section 9 by comparison with the K s 1 case; see the remarks after Lemma 9.1. Proposition 5.1 has the following consequence. COROLLARY 5.1. The interface process ␰ Ž⭈. defined by Ž5.8. satisfies a strong law of large numbers: for any y g R and t ) 0,

Ž 5.25.

lim nª⬁

1 n

␰ Ž w ny x , nt . s tg K Ž yrt . , ⺠-a.s.

PROOF. The proof has two parts. We do the limsup part and leave the liminf part to the reader. Let ␧ ) 0. By strict monotonicity we can pick ␦ ) 0 so that

␥ K Ž tg K Ž yrt . q ␧ , y . G ␥ K Ž tg K Ž yrt . , y . q ␦ G t q ␦ . The last inequality is actually an equality if Ž tg K Ž yrt ., y . is not a boundary point of UK . Let x s tg K Ž yrt . q ␧ . Almost surely the inequality TK Ž w nx x , w ny x . G n␥ K Ž x , y . y n ␦r2 G n Ž t q ␦r2 . holds eventually, and this implies by Ž5.8. that

␰ Ž w ny x , nt . F w nx x F ntg K Ž yrt . q n ␧ . Since ␧ ) 0 was arbitrary, lim sup nª⬁

1 n

␰ Ž w ny x , nt . F tg K Ž yrt . , ⺠-a.s.

I

Next we deduce a one-sided exponential bound valid for both the basic setting and the disordered setting. It is needed for Theorem 3 and for the strong law part of Theorem 1. Up to now, this section has required only ergodicity of Ž ␣ Ž j .., but this lemma requires a mixing assumption. So for convenience we take ␣ Ž j . i.i.d. as assumed in Section 3. LEMMA 5.4.

Ž 5.26.

Fix Ž x, y . g UK and ␧ ) 0. For some constant C ) 0, ⺠ Ž TK Ž w nx x , w ny x . F n␥ K Ž x , y . y n ␧ . F eyC n

for large enough n.

381


PROOF. Suppose first that y ) 0. Since ny1 ⺕w TK Žw nx x , w ny x.x ª ␥ K Ž x, y ., we may pick a number M such that ⺕ TK Ž w Mx x , w My x . G M Ž ␥ K Ž x , y . y ␧r2 . .

Ž 5.27. Let

Ž 5.28.

␶ l s TK Ž Ž l w Mx x , l w My x . , Ž Ž l q 1 . w Mx x , Ž l q 1 . w My x . . , l s 0, 1, 2, . . . .

Consider the mutual dependencies of the passage times ␶ l 4 under the measure ⺠. By definition Ž5.5. and the definition of K-admissibility, ␶ l is computed from

␶ Ž i , j . : l w Mx x q 1 F i F Ž l q 1. w Mx x , l w My x y w Mx x rK F j F Ž l q 1 . w My x q w Mx x rK 4 . The variables Ž ␭Ž i, j .. induce no dependencies between the ␶ l ’s, and the ␣ Ž j .’s force only a finite range dependence on the ␶ l ’s. Thus there is a number R such that ␶ lqj R : j G 04 are i.i.d. random variables, for any l G 0. L Ry1 Let L s w nrMR x . Then Ý ls0 ␶ l F TK Žw nx x , w ny x. . Abbreviate ␥ s ␥ K Ž x, y . and let ␤ ) 0. We apply standard large deviations reasoning, 1 n

Ž 5.29.

log ⺠ Ž TK Ž w nx x , w ny x . F n Ž ␥ y ␧ . . F F

LRy1

1 n 1 n

log ⺠

␶ l F nŽ ␥ y ␧ .

žÝ

ls0

/

ž ½

log exp Ž ␤ n Ž ␥ y ␧ . . ⺕ exp y␤

LRy1

Ý ls0

␶l

5/

.

To the expectation, apply first Holder’s inequality and then independence, ¨

ž ½

LRy1

⺕ exp y␤

Ý

␶l

ls0

5/

Ry1

s⺕

ž ½ ž ½ Ł exp

Ly1

y␤

ls0

F ⺕ exp y␤ R

Ý

␶ lqj R

js0 Ly1

Ý

␶jR

js0 L

s ⺕ Ž exp y␤ R␶ 0 4 . .

5/

5/


382

w Note to the reader: The above step is where more than ergodicity of Ž ␣ Ž i .. is needed.x Substitute this back into the last line of Ž5.29.: 1 n

log ⺠ Ž TK Ž w nx x , w ny x . F n Ž ␥ y ␧ . . F ␤ Ž␥ y ␧ . q

L

log ⺕ Ž exp Ž y␤ R␶ 0 . . n 1 ␧ F ␤ Ž␥ y ␧ . q 1y log ⺕ Ž exp Ž y␤ R␶ 0 . . MR 4␥

ž

/

' HŽ ␤ . . The last inequality is valid for large enough n. It remains to show that H Ž ␤ . - 0 for some ␤ ) 0. Since H Ž0. s 0, it suffices to show that H⬘Ž0. - 0. H⬘ Ž ␤ . s ␥ y ␧ y

1 M

ž

1y

␧ 4␥

/

⺕ Ž ␶ 0 exp Ž y␤ R␶ 0 . . ⺕ Ž exp Ž y␤ R␶ 0 . .

,

from which H⬘ Ž 0 . s ␥ y ␧ y

1 M

ž

1y

␧ 4␥

⺕ ␶0. .

/Ž

This is less than 0, for by Ž5.27. and Ž5.28., ⺕ Ž ␶ 0 . s ⺕ TK Ž w Mx x , w My x . G M Ž ␥ y ␧r2 . . This proves Ž5.26. for y ) 0. The argument for y - 0 is the same. For y s 0, choose by continuity x⬘ - x and y - 0 such that ␥ K Ž x⬘, y . G ␥ K Ž x, 0. y ␧r2. Since Žw nx x , 0. g Žw nx⬘x , w ny x. q LK for large n, we have TK Žw nx⬘x , w ny x. F TK Žw nx x , 0.. By the case already proved, ⺠ Ž TK Ž w nx x , 0 . F n Ž ␥ K Ž x , 0 . y ␧ . . F ⺠ Ž TK Ž w nx⬘ x , w ny x . F n Ž ␥ K Ž x⬘, y . y ␧r2 . . F eyC n .

I

The reader not interested in the disordered setting can move on to Section 6. For the disordered model, we need to think about the quenched setting. Proposition 5.1 implies that the limit Ž5.9. holds in ⺠ ␣ -probability, for Q-almost every ␣ . However, our proof of the hydrodynamic limit needs a stronger version where the quenched variable ␣ is translated simultaneously as the limit is taken. Write ␪ Ž k . for translations on the space A: for ␣ s Ž ␣ Ž i .., w ␪ Ž k . ␣ xŽ i . s ␣ Ž i q k .. PROPOSITION 5.2. There exists a subset A0 : A such that this holds: QŽ A0 . s 1, and for all ␣ g A0 , b g R, Ž x, y . g UK and ␧ ) 0,

Ž 5.30.

lim ⺠ ␪ Žw n b x.␣ Ž ny1 TK Ž w nx x , w ny x . y ␥ K Ž x , y . G ␧ . s 0.

nª⬁

383


PROOF. Start by observing this: the ⺠ ␪ Žw n b x.␣ -distribution of TK Ž w nx x , w ny x . s the ⺠ ␣ -distribution of TK Ž w nx x , w ny x . (⌰ Ž 0, w nb x .

Ž 5.31.

s the ⺠ ␣ -distribution of TK Ž w nx x , w ny x . (⌰ Ž w na x , w nb x . for any a g R. The second equality follows because the variables Ž ␭Ž i, j .. are i.i.d. ExpŽ1. under any ⺠ ␣ , so they can be translated arbitrarily with no effect on the distribution. Recall from Ž5.3. that only the second index l of the joint translation ⌰Ž k, l . affects ␣ . Fix b g R and Ž x, y . g UK . Since TK Žw nx x , w ny x.( ⌰ Ž0, w nb x. sd TK Žw nx x , w ny x. under ⺠ w sd denotes equality in distributionx , Lemma 5.4 implies that for all ␧ ) 0, ⬁

Ž 5.32.

Ý ⺠ Ž ny1 TK Ž w nx x , w ny x . (⌰ Ž 0, w nb x . F ␥ K Ž x , y . y ␧ . - ⬁. ns1

Consequently, by the Borel᎐Cantelli lemma, lim inf nª⬁

1 n

TK Ž w nx x , w ny x . (⌰ Ž 0, w nb x . G ␥ K Ž x , y .

⺠-a.s.,

and hence also ⺠ ␣ -a.s., for Q-a.e. ␣ . By Ž5.31., for Q-a.e. ␣ ,

Ž 5.33.

lim ⺠ ␪ Žw n b x.␣ Ž ny1 TK Ž w nx x , w ny x . F ␥ K Ž x , y . y ␧ . s 0.

nª⬁

At this point we have an exceptional Q-null set of ␣ ’s that depends on b, Ž x, y . and ␧ . To get a single null set simultaneously valid for all b, Ž x, y . and ␧ requires an approximation step and use of the continuity of ␥ K as in the proof of Proposition 5.1. We leave this to the reader. Equation Ž5.33. is one half of the goal Ž5.30.. To get the remaining half, define the event ⌺ 1 : ⌺ as the set of Ž ␭ , ␣ . for which both of these statements hold:

Ž 5.34.

lim nª⬁

1 n

TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . s ␥ K Ž x , y .

for all Ž x , y . g UK

and

Ž 5.35.

lim nª⬁

1 n

TK Ž Ž w nu x , w nv x . , Ž 0, 0 . . s ␥ K Ž yu, yv . UK . for all Ž u, v . g yU

Statement Ž5.34. is of course just Proposition 5.1, and Ž5.35. is the same thing with a change in lattice direction. So ⺠Ž ⌺ 1 . s 1. Define the ‘‘good’’ subset B : A as the set of ␣ such that ⺠ ␣ Ž ⌺ 1 . s 1. Then QŽ B. s 1. We shall


384

demonstrate that there always is a number a, depending on b, Ž x, y . and ␧ , such that lim ⺠ ␣ Ž ny1 TK Ž w nx x , w ny x . (⌰ Ž w na x , w nb x .

nª⬁

Ž 5.36.

G ␥K Ž x , y . q ␧ . s 0

for ␣ g B. This step is carried out in several cases according to the values of b and Ž x, y .. Then we are done, for another application of Ž5.31. shows that Ž5.36. and Ž5.33. together give Ž5.30.. CASE I Ž b, y ) 0 or b, y - 0.. Set a s bxry ) 0. By superadditivity, TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . (⌰ Ž w na x , w nb x . s TK Ž Ž w na x , w nb x . , Ž w na x q w nx x , w nb x q w ny x . .

Ž 5.37.

F TK Ž Ž 0, 0 . , Ž w na x q w nx x , w nb x q w ny x . . y TK Ž Ž 0, 0 . , Ž w na x , w nb x . . ,

from which lim sup nª⬁

Ž 5.38.

1 n

TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . (⌰ Ž w na x , w nb x .

F ␥ K Ž a q x , b q y . y ␥ K Ž a, b . s Ž 1 q bry . ␥ K Ž x , y . y Ž bry . ␥ K Ž x , y . s ␥K Ž x , y .

on the event ⌺ 1. The second-last step used homogeneity Ž5.18.. It follows that Ž5.36. holds for all ␣ g B in Case I. For the next case we need an intermediate lemma. LEMMA 5.5.

Ž 5.39.

For x ) 0 and b g R, lim ␥ K Ž a q x , b . y ␥ K Ž a, b . s ␥ K Ž x , 0 . .

aª⬁

PROOF. Superadditivity says that the liminf of the left-hand side of Ž5.39. is at least as large as the right-hand side. The nontrivial direction is the opposite inequality. The case b s 0 follows from homogeneity. Suppose b ) 0. Set x⬘ s

xŽ a q x. a q x q Kb

g Ž 0, x . .


385

Check from a picture that Ž x, 0. g Ž x⬘, bx⬘rŽ a q x .. q UK . From this we get the first inequality below, and then we use homogeneity.

␥ K Ž x , 0 . G ␥ x⬘,

ž

s

x⬘ aqx

bx⬘ aqx

/

␥K Ž a q x , b .

s ␥K Ž a q x , b . y 1 q

ž

x y x⬘ a

G ␥ K Ž a q x , b . y ␥ K Ž a, b . y

␥ K a,

ab

/ ž / ž / Ž aqx

x y x⬘ a

␥ K a, b .

because ␥ K Ž a, abr Ž a q x . . F ␥ K Ž a, b . for a, b, x ) 0 G ␥ K Ž a q x , b . y ␥ K Ž a, b . y O Ž ay1 . , because x y x⬘ s O Ž ay1 . and because the moment bound Ž5.10. implies that ␥ K Ž a, b . s O Ž a. for a fixed b. This proves Ž5.39. for b ) 0. We leave the similar argument for b - 0 to the reader. I We return to the proof of Proposition 5.2. CASE II Ž y s 0.. Pick a large enough so that

␥ K Ž a q x , b . y ␥ K Ž a, b . F ␥ K Ž x , 0 . q ␧r2.

Ž 5.40.

Repeat the reasoning of Case I down to the second line of Ž5.38. with y s 0. Then Ž5.40. gives lim sup nª⬁

1 n

TK Ž Ž 0, 0 . , Ž w nx x , 0 . . (⌰ Ž w na x , w nb x . F ␥ K Ž x , 0 . q ␧r2

everywhere on the event ⌺ 1. Again Ž5.36. holds for all ␣ g B. CASE III Ž b ) 0 ) y G yb or b - 0 - y F yb .. If y s yb we can directly apply Ž5.35. to lim nª⬁

1 n

TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . (⌰ Ž w ynx x , w nb x .

s lim

nª⬁

s lim

nª⬁

1 n 1 n

TK Ž Ž w ynx x , w yny x . , Ž w nx x q w ynx x , w ny x q w yny x . . TK Ž Ž w ynx x , w yny x . , Ž 0, 0 . . .

The point of the last equality is simply that, due to the continuity of ␥ K , the difference between Žw nx x q w ynx x , w ny x q w yny x. and Ž0, 0. is not felt in the limit.


386

UK . By Otherwise set a s bxry - 0, and we have Ž a, b ., Ž a q x, b q y . g yU superadditivity, TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . (⌰ Ž w na x , w nb x .

Ž 5.41.

s TK Ž Ž w na x , w nb x . , Ž w na x q w nx x , w nb x q w ny x . . F TK Ž Ž w na x , w nb x . , Ž 0, 0 . . y TK Ž Ž w na x q w nx x , w nb x q w ny x . , Ž 0, 0 . . ,

from which by Ž5.35., lim sup nª⬁

1 n

TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . (⌰ Ž w na x , w nb x .

F ␥ K Ž ya, yb . y ␥ K Ž ya y x , yb y y .

Ž 5.42.

s y

b

ž / y

␥K Ž x , y . y y

ž

b y

y 1 ␥K Ž x , y .

/

s ␥K Ž x , y . on the event ⌺ 1. The assumption < y < - < b < was needed for the homogeneity step, so that ybry y 1 ) 0. Ž5.36. holds for all ␣ g B also in this case. CASE IV Ž b ) 0 and yb ) y .. Pick a finite partition x 0 , x 1 , . . . , x m 4 of the interval Ž0, x . such that x 0 - Kb - x 1 - ⭈⭈⭈ - x my1 - x q K Ž b q y . - x m . The point here is that any K-admissible path from Ž1, w nb x. to Žw nx x , w nb x q w ny x. must cross the x-axis at one of the sites Ž i, 0. for w nx 0 x F i F w nx m x . By uniform continuity of ␥ K on the compact set w Kb, x x = yb4 , we may further choose the above partition fine enough to have

Ž 5.43.

␥ K Ž x j , yb . y ␥ K Ž x jq1 , yb . F ␧r2 for all j.

Here and below, interpret ␥ K Ž x 0 , yb . as ␥ K Ž Kb, yb . because ␥ K Ž x 0 , yb . is not defined for x 0 - Kb. Now we can estimate TK Ž Ž 0, 0 . , Ž w nx x , w ny x . . (⌰ Ž 0, w nb x . s TK Ž Ž 0, w nb x . , Ž w nx x , w nb x q w ny x . .

Ž 5.44.

F

max 0FjFmy1

½ T Ž Ž 0, w nb x . , Ž w nx K

jq1

x , 0. .

qTK Ž Ž w nx j x , 0 . , Ž w nx x , w nb x q w ny x . . .

5

By superadditivity of ␥ K and Ž5.43.,

Ž 5.45.

␥ K Ž x , y . q ␧ G ␥ K Ž x j , yb . q ␥ K Ž x y x j , b q y . q ␧ G ␥ K Ž x jq1 , yb . q ␥ K Ž x y x j , b q y . q ␧r2.

387


Combining Ž5.44. and Ž5.45. allows us to write ⺠ ␣ TK Ž w nx x , w ny x . (⌰ Ž 0, w nb x . G n␥ K Ž x , y . q n ␧ 4 my1

F

Ý js0

⺠ ␣ TK Ž Ž 0, w nb x . , Ž w nx jq1 x , 0 . . G n␥ K Ž x jq1 , yb . q n ␧r4

½

5

q⺠ ␣ TK Ž Ž w nx j x , 0 . , Ž w nx x , w nb x q w ny x . .

½

G n␥ K Ž x y x j , b q y . q n ␧r4 my1

s

Ý js0

5

⺠ ␣ TK Ž Ž 0, 0 . , Ž w nx jq1 x , y w nb x . . (⌰ Ž 0, w nb x .

½

G n␥ K Ž x jq1 , yb . q n ␧r4

5

q⺠ ␣ TK Ž Ž 0, 0 . , Ž w nx x y w nx j x , w nb x q w ny x . .

½

G n␥ K Ž x y x j , b q y . q n ␧r4

5

.

In the last step we do a trivial translation in the first probability inside the brackets, and a translation that utilizes the horizontal translation-invariance of Ž ␭Ž i, j .. in the second probability. The last sum tends to 0 as n ª ⬁, because Case III takes care of the first probability, and assumption Ž5.34. of the second. CASE V Ž b - 0 and yb - y .. This is the only remaining case. Perform a partition argument similar to the one for Case IV and appeal to earlier cases. I 6. Proof of the weak laws. In this section we prove Theorem 3. Simultaneously we obtain statement Ž2.13. of Theorem 1 under assumption Ž2.4.. w This is just the special case ␣ Ž i . ' 1 of Theorem 3.x Fix a configuration ␣ g A0 , the set defined by Proposition 5.2. Let Ž␩nŽ i, 0.: i g Z. be the given random initial configurations that satisfy Ž3.2.. To define initial configurations for the corresponding server processes, set z n Ž 0, 0 . s 0, i

Ž 6.1.

z n Ž i , 0. s

Ý ␩n Ž j, 0.

for i ) 0, and

js1 0

z n Ž i , 0. s y

Ý

␩n Ž j, 0 .

for i - 0.

jsiq1

Let z nŽ⭈. denote the server process constructed from initial configuration Ž z nŽ i, 0.: i g Z. according to the description of Section 4, and denote its probability measure by Pn␣ . Then define the K-exclusion ␩nŽ⭈. by Ž4.1..


388

The empirical measure of an interval Ž a, b x for a - b is given by

␲n Ž t , Ž a, b x . s s

Ž 6.2.

s

1 n 1 n 1 n

Ý ␩n Ž j, t . IŽ a , b x

jgZ

j

ž / n

w nb x

␩n Ž j, t .

Ý

js w na x q1

z n Ž w nb x , t . y

1 n

z n Ž w na x , t . .

Pick U0 that satisfies Ž2.7. and U0 Ž0. s 0. Then assumption Ž3.2. together with Ž6.1. and Ž6.2. implies that, for all y g R and ␧ ) 0,

Ž 6.3.

lim Pn␣ Ž ny1 z n Ž w ny x , 0 . y U0 Ž y . G ␧ . s 0.

nª⬁

Note that behind this claim is the fact that the limit measure u 0 Ž x . dx of ␲nŽ0, dx . has no atoms, so that ␲nŽ0, Ž a, b x. converges to U0 Ž b . y U0 Ž a. for all intervals Ž a, b x . Define UŽ x, t . by Ž2.8.. By Ž6.2. and the definition Ž2.9. of uŽ x, t ., we shall have proved Theorem 3 if we show that, for all x g R, t ) 0, and ␧ ) 0,

Ž 6.4.

lim Pn␣ Ž ny1 z n Ž w nx x , nt . y U Ž x , t . G ␧ . s 0.

nª⬁

Fix Ž x, t .. The starting point is the coupling Ž4.5. and Ž4.9. that now reads z n Ž w nx x , nt . s sup w z n Ž i , 0. Ž w nx x y i , nt . igZ

Ž 6.5.

s sup z n Ž i , 0 . y ␰ z nŽ i , 0. Ž w nx x y i , nt . 4 . igZ

An important point to keep straight is that, by the same reasoning that justified Lemma 5.1, we have this equality in distribution:

Ž 6.6.

d

␰ z n Žw n b x , 0. Ž ⭈ . under Pn␣ s ␰ Ž ⭈ . under ⺠ ␪ Žw n b x.␣ .

The first task is to restrict the range of indices i that need to be considered in Ž6.5.. For r 1 - r 2 , define

Ž 6.7.

␨n Ž r 1 , r 2 . s

max

w nr 1 x FiF w nr 2 x

z n Ž i , 0. y ␰ z Ž i , 0. Ž w nx x y i , nt . 4 . n

LEMMA 6.1. For any ␣ g A and r 1 - x y t - x q t - r 2 , there exists a constant C ) 0 such that

Ž 6.8.

Pn␣ Ž z n Ž w nx x , nt . / ␨n Ž r 1 , r 2 . . F eyC n

for all n. PROOF. By Lemma 4.4 it suffices to show that Pn␣ Ž ␰ z n Žw n r 1 x , 0. Ž w nx x y w nr 1 x , nt . ) 0 . F eyC n


389

and Pn␣ Ž ␰ z nŽw n r 2 x , 0. Ž w nx x y w nr 2 x , nt . ) K Ž w nr 2 x y w nx x . . F eyC n . By Ž6.6., this will follow from showing that, for any b g R and y ) t, ⺠ ␪ Žw n b x.␣ Ž ␰ Ž w ny x , nt . ) 0 . F eyC n

Ž 6.9. and

⺠ ␪ Žw n b x.␣ Ž ␰ Ž y w ny x , nt . ) K w ny x . F eyC n .

Ž 6.10.

We show Ž6.9. and leave the argument for Ž6.10. to the reader. By definitions Ž5.5. and Ž5.8., and because ␣ Ž j . F 1 always,

␰ Ž w ny x , nt . (⌰ Ž 0, w nb x . ) 0 « TK Ž 1, w ny x . (⌰ Ž 0, w nb x . F nt w nb x q w ny x

«

Ý

␣ Ž j.

y1

␭Ž 1, j . F nt

js w nb x w nb x q w ny x

«

Ý

␭Ž 1, j . F nt.

js w nb x

This last event has exponentially small probability because the Ž ␭Ž1, j .. are i.i.d. with expectation 1 and y ) t. I The second point is to note the weak law of the interface processes in Ž6.5.. This is a corollary of Proposition 5.2. LEMMA 6.2.

Ž 6.11.

For any ␣ g A0 , b, y g R, t ) 0 and ␧ ) 0,

lim Pn␣ Ž ny1 ␰ z n Žw n b x , 0. Ž w ny x , nt . y tg K Ž yrt . G ␧ . s 0.

nª⬁

PROOF. The lemma follows from Ž6.6. and Proposition 5.2, by the same sort of reasoning that Corollary 5.1 followed from Proposition 5.1. I One half of Ž6.4., namely,

Ž 6.12.

lim Pn␣ Ž ny1 z n Ž w nx x , nt . F U Ž x , t . y ␧ . s 0,

nª⬁

is now immediate because setting i s w ny x inside the braces in Ž6.5. gives a lower bound for z nŽw nx x , nt ., and by Ž2.8., Ž6.3. and Ž6.11., the limit of the quantity in braces can be taken arbitrarily close to UŽ x, t . by choice of y. For the other half, fix r 1 - r 2 so that Ž6.8. holds, and pick a partition r 1 s b 0 - b 1 - ⭈⭈⭈ - bmy1 - bm s r 2 fine enough so that

Ž 6.13.

tg K Ž Ž x y b lq1 . rt . y tg K Ž Ž x y b l . rt . F ␧r2 for all l.


390

By Lemma 4.3 we may reason as follows:

␨n Ž r 1 , r 2 . s

max

w nr 1 x FiF w nr 2 x

s max

w z n Ž i , 0. Ž w nx x y i , nt .

max

0Fl-m w nb l x FiF w nb lq1 x

Ž 6.14.

F max

0Fl-m

w z n Ž i , 0. Ž w nx x y i , nt .

w z Žw n b x , 0. Ž w nx x y w nbl x , nt . n

l

qz n Ž w nblq1 x , 0 . y z n Ž w nbl x , 0 . 4 s max

0Fl-m

z n Ž w nblq1 x , 0. y ␰ z Žw n b x , 0. Ž w nx x y w nbl x , nt . 4 . n

l

By Ž6.3. and Ž6.11., the limit in Pn␣ -probability of Ž1rn. ⭈ w the last line of Ž6.14.x is max U0 Ž b lq1 . y tg K Ž Ž x y b l . rt . 4 , 0Fl-m

which by Ž6.13., and by the definition Ž2.8. of UŽ x, t ., is bounded above by UŽ x, t . q ␧r2. We have shown that lim Pn␣ Ž ny1␨n Ž r 1 , r 2 . G U Ž x , t . q ␧ . s 0,

Ž 6.15.

nª⬁

which together with Ž6.8. and Ž6.12. implies Ž6.4., the goal of the argument. Theorem 3 and the weak law part of Theorem 1 are thereby proved. As a final matter in this section, we can now see why we have to bound the rates ␣ Ž j . away from zero in order to get nontrivial hydrodynamics. Suppose QŽ ␣ Ž0. - ␧ . ) 0 for all ␧ ) 0. Then the limit in Proposition 5.1 would be ␥ K Ž x, y . ' ⬁. For y ) 0 this can be seen from 1 n

TK Ž w nx x , w ny x . G

½

min

0FjF w ny x

␣ Ž j.

5

y1

1

w nx x

n

is1

Ý ␭Ž i , jn . ,

where the random jn satisfies ␣ Ž jn . s min 0 F j F w n y x ␣ Ž j .. For y F 0, similar reasoning works. From ␥ K ' ⬁ follows that g K Ž y . s 0 for y G 0 and g K Ž y . s yKy for y F 0. This in turn makes UŽ x, t . s U0 Ž x . in Ž2.8.. 7. An upper tail estimate for the last-passage times. To obtain the strong laws in Theorems 1 and 2, we need estimates that allow us to apply the Borel᎐Cantelli lemma. In Lemma 5.4 we already have a lower tail estimate. This section is devoted to the next lemma that gives the complementary upper tail estimate. Its proof is reminiscent of a proof of Grimmett and Kesten Ž1984. for a lower tail estimate on the passage times of firstpassage percolation, although the main challenge here is different. The geometry of our paths is simpler than in first-passage percolation. What makes our upper tail estimate problematic is that passage times of individual sites are unbounded, so an excessively high value of TK Žw nx x , w ny x. can be produced in a large number of ways. In contrast, a lower tail estimate can use the fact that passage times are bounded below by zero. In the end we do not quite get an exponential bound, although we certainly expect one, by analogy Ž1998b, c.x . with other similar models w Seppalainen ¨¨

391


LEMMA 7.1. Suppose a0 s ␣ Ž j . s 1 for all j g Z. Fix Ž x, y . g UK and ␧ ) 0. Then there is a constant C ) 0 such that

Ž 7.1.

⺠ Ž TK Ž w nx x , w ny x . G n␥ K Ž x , y . q n ␧ . F exp yCn Ž log n .

y1

for large enough n. The proof of this lemma takes several steps. Regard Ž x, y . fixed for the duration of this section. We start with an exponential upper tail bound on large enough deviations of order n. LEMMA 7.2. Suppose a0 s ␣ Ž j . s 1 for all j g Z. For Ž x, y . g UK , there exist positive constants b 0 , n 0 and c 0 depending on Ž x, y . such that

Ž 7.2.

⺠ Ž TK Ž w nx x , w ny x . G bn . F exp Ž yc0 bn .

for all b G b 0 and n G n 0 . PROOF. By Ž5.11., Ž5.13. and Lemma 5.2, it suffices to prove the bound for the sum S 1r2 Ž2w nx x q w ny x. of 2w nx x q w ny x i.i.d. ExpŽ1r2.-distributed random variables. The conclusion then follows from a standard large deviation bound, Prob Ž S 1r2 Ž 2 w nx x q w ny x . G bn .

Ž 7.3.

F exp y Ž 2 w nx x q w ny x . I Ž bn Ž 2 w nx x q w ny x .

½

y1

.5,

where

Ž 7.4.

I Ž x . s xr2 y 1 y log Ž xr2 .

is the Cramer ´ rate function for the ExpŽ1r2. distribution. The reader can prove Ž7.3. easily with the exponential Chebyshev argument of Lemma 5.4 above. Alternatively, see Section 2.2 in Dembo and Zeitouni Ž1993., Section 1.2 in Deuschel and Stroock Ž1989., Section 1.9 in Durrett Ž1991. or Section 3 in Varadhan Ž1984.. The inequality Ž7.3. is true already for finite n due to superadditivity. By Ž7.4., I Ž bn Ž 2 w nx x q w ny x .

y1

.G

1 4

bn Ž 2 w nx x q w ny x .

y1

for large enough b, so the bound expw yc0 bn x follows. I Consider the level curve Ž x, y . g UK : ␥ K Ž x, y . s ␥ K Ž x, y .4 . Since ␥ K is concave and strictly increasing in both variables, this level curve is the graph of a convex function y s hŽ x .. Pick a tangent line of the curve at Ž x, y .. This line has slope y␳ g Žy⬁, y1rK .. ŽThe slope cannot be equal to y⬁ or y1rK because the level curve connects a point on the positive y-axis to a point on the positive part of the line y s yxrK.. The tangent line itself intersects the y-axis and the line y s yxrK at the points Ž0, y 1 . and Ž Ky 2 , yy 2 . for y 1 s y q ␳ x and y 2 s Ž y q ␳ x .rŽ K ␳ y 1.. Set y 0 s Ž y 1 q y 2 .r2.


392

On the lattice Z 2 define, for large M g N, BM s Ž i , j . g Z 2 : y w ␳ i x F j F w My1 x y w ␳ i x y 1, and for

Ž 7.5.

some 0 F k F w My 0r␳ x , there is a K-admissible path from

Ž k , y w ␳ k x . to Ž i , j . 4 . The set BM is the lattice analogue, in the scale 1rM, of the trapezoid between the lines y s y 1 y ␳ x and y s y␳ x, with vertices Žclockwise. Ž0, 0., Ž0, y 1 ., Ž3 y 0r␳ , yy 0 y y 2 . and Ž y 0r␳ , yy 0 .. Let F be the closure of this trapezoid on the plane. See Figure 1. Recall the definition Ž5.4. of the passage time T Ž r . of a K-admissible path. Let

Ž 7.6.

Y Ž BM . s max T Ž r . : r is a K-admissible path : BM 4 .

LEMMA 7.3. The limit 1

Y Ž BM . s ␥ K Ž x , y . M holds ⺠-a.s. There exist positive constants b 0 , M0 and c 0 such that

Ž 7.7. Ž 7.8.

lim

Mª⬁

⺠ Ž Y Ž BM . G Mb . F exp w yc0 Mb x

for all b G b 0 and M G M0 .

FIG. 1. The region UK is bounded by the y-axis and the line y s yxrK Ž dashed lines . that meet at the origin O. A s Ž x, y . on the level curve ␥ K s ␥ K Ž x, y .4. BD is the tangent line at A s Ž x, y . of slope y␳ . It intersects the boundary of UK at the points B s Ž0, y1 . and C s Ž Ky 2 , yy 2 .. OE is a line of slope y␳ through the origin, ED a line of slope y1rK. The vertices of the trapezoid F are OBDE.

393


PROOF. Since

w My x s My1 y M ␳ x F w My1 x y w M ␳ x x ,

Ž 7.9.

every K-admissible path from Ž0, 1. to Žw Mx x , w My x y 1. lies inside BM , and consequently by the continuity of ␥ K , lim inf Mª⬁

1 M

Y Ž BM . G ␥ K Ž x , y . , ⺠-a.s.

For the converse, suppose a small number ␦ 1 ) 0 is given. Then by the compactness of F it is possible to pick two even smaller numbers ␦ , ␧ ) 0 and two finite sequences xX0 - xX1 - ⭈⭈⭈ - xXI

and

xY0 - xY1 - ⭈⭈⭈ - xYJ

such that this holds: with yXk s y␧ y ␳ xXk and yYl s y 1 q ␧ y ␳ xYl , we have the inclusions

Ž 7.10. F Ž ␦ . :

I

D Ž xXk , yXk . q UK

and

ks0

F Ž␦ . :

J

D Ž xYl , yYl . y UK

,

ls0

where F Ž ␦ . is the ␦-neighborhood of F, and moreover, all Ž xXk , yXk . and Ž xYl , yYl . lie in F Ž ␦ 1 .. See Figure 2. Let I be the set of pairs Ž k, l . such that Ž xYl , yYl . g Ž xXk , yXk . q UK . It then follows, for large enough M, that if r is any K-admissible path in BM , there is a pair Ž k, l . g I and a K-admissible path r ⬘ = r from Žw MxXk x q 1, w MyXk x. to Žw MxYl x , w MyYl x. . Consequently, Y Ž BM . F max TK Ž Ž w MxXk x , w MyXk x . , Ž w MxYl x , w MyYl x . . ,

Ž 7.11.

Ž k , l .g I

and then lim sup Mª⬁

1 M

Y Ž BM . F max ␥ K Ž xY2 y xXk , yY2 y yXk . , ⺠-a.s. Ž k , l .g I

To establish Ž7.7. it remains to argue that, given ␧ 0 , we can make max ␥ K Ž xYl y xXk , yYl y yXk . F ␥ K Ž x , y . q ␧ 0

Ž k , l .g I

by choosing ␦ 1 small enough. Consider any point Ž x⬘, y␳ x⬘., x⬘ g w 0, y 0r␳ x , on the southwest edge of F Žedge OE in Figure 1.. By convexity and the choice of the slope ␳ , the translated level curve

Ž x , y . g Ž x⬘, y␳ x⬘. q UK : ␥ K Ž x y x⬘, y q ␳ x⬘. s ␥ K Ž x , y . 4 is on or above the line y s y 1 y ␳ x Žline BD in Figure 1., and consequently ␥ K Ž x y x⬘, y q ␳ x⬘. F ␥ K Ž x, y . for all Ž x, y . g wŽ x⬘, y␳ x⬘. q UK x l F. This works for all points on the southwest edge of F. Therefore sup ␥ K Ž x y x⬘, y y y⬘ . : Ž x , y . , Ž x⬘, y⬘ . g F , Ž x , y . g Ž x⬘, y⬘ . q UK 4 s ␥K Ž x , y . .


394

X X FIG. 2. w To justify Ž7.10.x . The trapezoid represents the neighborhood F Ž ␦ ., A s Ž x k , yk ., and Y Y X X B s Ž x l , yl .. The dash-dot line bounds the set Ž x k , yk . q UK . The reader sees that finitely many such sets can cover F Ž ␦ ., however close to F Ž ␦ . we require the basepoints A to be. The same Y Y reasoning applies to Ž x l , yl . y UK bounded by the dashed line.

By continuity of ␥ K and compactness of F, for small enough ␦ 1 ) 0, we have sup ␥ K Ž x y x⬘, y y y⬘ . : Ž x , y . , Ž x⬘, y⬘ . g F Ž ␦ 1 . , Ž x , y . g Ž x⬘, y⬘ . q UK 4 F ␥K Ž x , y . q ␧ 0 . This completes the proof of Ž7.7.. The deviation bound Ž7.8. follows by applying Lemma 7.2 to the finitely many random variation on the right-hand side of Ž7.11.. I The preliminaries are done and we can attack Lemma 7.1 PROOF OF LEMMA 7.1. Abbreviate ␥ s ␥ K Ž x, y .. Suppose M is large enough so that Ž7.8. is valid, and let n 4 M. Recall the definition of the number y 1 in the paragraph preceding Ž7.5. Žpoint B in Figure 1.. Let R 0 be the smallest integer that satisfies

Ž 7.12.

R 0 w My1 x G w ny1 x q 1.

Given ␧ 1 ) 0,

Ž 7.13.

Ž 1 y ␧ 1 . nrM F R 0 F Ž 1 q ␧ 1 . nrM


395

for large enough n. Divide the lattice LK into parallel strips of slope y␳ and height w My1 x , so that the k th strip, 1 F k F R 0 , is given by

Ž 7.14.

Tk s Ž i , j . g LK : Ž k y 1 . w My1 x y w ␳ i x F j F k w My1 x y w ␳ i x y 1 4 .

Equations Ž7.9. and Ž7.12. guarantee that Žw nx x , w ny x. lies in the union T1 j ⭈⭈⭈ j TR 0 . The definition Ž7.5. of BM was tailored so that the height of BM matches the height of a strip Tk . We can cover each strip with finitely many translates of the set BM so that these translates do not intersect on the bottom boundary line of the strip. That is, if B⬘, B⬙ are two such translates inside Tk and Ž i, Ž k y 1.w My1 x y w ␳ i x. g B⬘, then Ž i, Ž k y 1.w My1 x y w ␳ i x. f B⬙. Fix such a covering of each strip Tk by translates of BM , henceforth called basic regions. For T1 , only one basic region is needed. If M is large enough, any K-admissible path that starts in a basic region B : Tk can enter only five possible basic regions inside Tkq 1. ŽFigure 3.. Thus there are at most 5 R 0

FIG. 3. The dashed lines bound the strips Tk and Tkq1 . Here B0 is a basic region in Tk , and B1 , . . . , B5 are basic regions in Tkq1 . The figure gives a macroscopic representation. For i s 1, . . . , 5, the label Bi is placed at the top vertex of the trapezoid that represents Bi . B1 , . . . , B5 overlap in the interior of Tkq 1 but not on the dashed boundary between the two strips. B0 can touch at most five basic regions in Tkq 1 due to our choice of y 0 s Ž y1 q y 2 .r2. This makes the length of the northeast edge Ž BD in Figure 1. of the trapezoid three times the length of the southwest edge Ž OE in Figure 1..


396

FIG. 4. An example of a sequence BŽ1., . . . , BŽ R 0 . of basic regions whose union contains K-admissible paths from Ž1, 0. to the northeast edge of BŽ R 0 .. The solid lines mark the boundaries of the strips Tk . The figure gives a macroscopic view so the basic regions are represented by shaded trapezoids. Here BŽ1. in T1 is reduced to a triangle because the trapezoid is larger than T1 .

sequences BŽ1., . . . , BŽ R 0 . of basic regions such that BŽ j . : Tj and such that the union of the BŽ j .’s can contain a K-admissible path from Ž1, 0. to Žw nx x , w ny x. . ŽFigure 4.. If BŽ1., . . . , BŽ R 0 . is such a sequence that contains the K-admissible path r, then T Ž r . G n␥ q n ␧ implies Y Ž BŽ1.. q ⭈⭈⭈ qY Ž BŽ R 0 .. G n␥ q n ␧ , where the random variable Y Ž BŽ j .. is defined as in Ž7.6.. Now fix a sequence BŽ1., . . . , BŽ R 0 .. We shall estimate the probability R0

⺠

ž

Ý Y Ž BŽ j . . G n␥ q n ␧ js1

/

.

Incidentally, an ordinary exponential large deviation estimate for the i.i.d. random variables Y Ž B Ž j .. of the type ⺠ ŽÝ 1R 0 Y Ž B Ž j .. G R 0 ␤ . F expw yR 0 QŽ ␤ .x for some unknown rate function Q would not be of help now because, in the final step, our estimate has to be multiplied by the factor 5 R 0 , to account for all the possible ways of choosing the sequence BŽ1., . . . , BŽ R 0 ..

397


Pick and fix a number ␤ 0 ) 0. In the first step we decompose like this: R0

ž

⺠

Ý Y Ž BŽ j . . G n␥ q n ␧ js1

/

R0

F

Ž 7.15.

Ý ⺠ Ž Y Ž BŽ j . . G ␤0 n . js1 R0

q⺠

ž

Ý Y Ž BŽ j . . G n␥ q n ␧ , Y Ž BŽ j . . - ␤ 0 n for 1 F j F R 0 js1

/

.

Next we partition Ž0, ␤ 0 n. and count the number of Y Ž BŽ j ..’s that fall into each partition interval. First, pick small ␦ ) 0 and ␧ 2 g Ž0, ␧ .. Let L0 s w b 0r␦ x q 1,

Ž 7.16.

where b 0 is the constant that appears in Lemma 7.3, and L1 s w ␤ 0 nrM␦ x q 1.

Ž 7.17. Set R0

␨0 s

Ý I Y Ž B Ž j . . - M Ž ␥ q ␧ 2 . 4 , js1 R0

␨L 0 s

Ý I M Ž ␥ q ␧ 2 . F Y Ž BŽ j . . - ML0 ␦ 4 js1

and R0

␨l s

Ý I M Ž l y 1. ␦ F Y Ž BŽ j . . - Ml ␦ 4

for L 0 - l F L1 .

js1

In the next step, apply Ž7.8. to the probabilities ⺠Ž Y Ž BŽ j .. G ␤ 0 n. in Ž7.15.. This is valid for nrM large enough. In the last probability in Ž7.15., pick ␧ 3 ) 0 and separate cases according to the value of ␨ L 0 . Then Ž7.15. becomes R0

⺠

ž

Ý Y Ž B Ž j . . G n␥ q n ␧ js1

/

F R 0 exp w yc0 ␤ 0 n x q ⺠ Ž ␨ L 0 ) ␧ 3 R 0 .

Ž 7.18.

R0

q⺠

ž

Ý Y Ž B Ž j . . G n␥ q n ␧ , js1

/

Y Ž B Ž j . . - ␤ 0 n for 1 F j F R 0 , ␨ L 0 F ␧ 3 R 0 .


398

In the last probability, the possible realizations of Ž ␨ 0 , ␨ L 0 , . . . , ␨ L 1 . are seL 1yL 0q2 quences w s Ž w 0 , wL 0 , . . . , wL 1 . g Zq of nonnegative integers that satisfy these properties: L1

w0 q

Ž 7.19.

Ý

wl s R 0 ,

lsL 0 L1

w0 M Ž ␥ q ␧ 2 . q

Ž 7.20.

Ý

wl Ml ␦ G n Ž ␥ q ␧ .

lsL 0

and wL 0 F ␧ 3 R 0 .

Ž 7.21. Let

L 1yL 0q2 G s w g Zq : w satisfies Ž 7.19. , Ž 7.20. and Ž 7.21. 4 .

The last probability in Ž7.18. is bounded above by ⺠ŽŽ ␨ 0 , ␨ L 0 , . . . , ␨ L 1 . g G .. Let us estimate the probability of a single element w g G . Abbreviate R0 R0 s w ,w ,...,w w 0 L0 L1

ž / ž

/

for the number of ways of arranging the Y Ž BŽ j ..’s in the partition intervals, with counts Ž w 0 , wL 0 , . . . , wL 1 .. Then, since the Y Ž BŽ j ..’s are i.i.d. and distributed like Y Ž BM ., ⺠ Ž Ž ␨ 0 , ␨ L 0 , . . . , ␨ L 1 . s Ž w 0 , wL 0 , . . . , wL 1 . . F

R0 w

ž / ž / ž /

L1

Ł

lsL 0q1

⺠ Ž M Ž l y 1 . ␦ F Y Ž BM . - Ml ␦ . L1

R0 F exp yc0 Ý w l M Ž l y 1 . ␦ w lsL q1 F

½ ½

0

wl

5

by Ž 7.8.

R0 exp yc0 n Ž ␥ q ␧ . y w 0 M Ž ␥ q ␧ 2 . y wL 0 ML0 ␦ w L1

y

Ý

w l M␦

lsL 0q1

F

5

by Ž 7.20.

R0 exp yc0 n Ž ␥ q ␧ . y R 0 M␦ y w 0 M Ž ␥ q ␧ 2 y ␦ . w

ž /

½

ywL 0 M Ž L0 y 1 . ␦

5

by Ž 7.19.

399


F

R0 exp yc0 n ␥ q ␧ y Ž 1 q ␧ 1 . Ž ␥ q ␧ 2 . y ␧ 3 Ž 1 q ␧ 1 . b 0 w

ž /

4

by Ž 7.13. , Ž 7.16. and Ž 7.21. and by choosing ␦ small enough F

R0 exp w yc1 n x , w

ž /

for a constant c1 ) 0, by choosing ␧ 1 , ␧ 2 and ␧ 3 small enough. w Note to the reader: The first step above would not work for the disordered case, because the Y Ž BŽ j ..’s might not be independent. Indeed, for some sequence BŽ1., . . . , BŽ R 0 ., the Y Ž BŽ j ..’s might all depend on some particular ␣ Ž j .. This can be fixed by conditioning on ␣ , but then the Y Ž BŽ j ..’s are no longer identically distributed. On account of these complications we state and prove Lemma 7.1 only for the basic case without disorder.x Let ␦ Ž M . s ⺠Ž Y Ž BM . G M Ž␥ q ␧ 2 ... By Ž7.7. we may suppose M is large enough to have ␦ Ž M . ␧ 3 - 1r2, for a fixed ␧ 3 ) 0. Returning to Ž7.18., we have R0

⺠

ž


/

F R 0 exp w yc0 ␤ 0 n x q ⺠ Ž ␨ L 0 ) ␧ 3 R 0 . q

Ý wg G

R0 exp w yc1 n x w

ž /

Ž 7.22. F R exp w yc ␤ n x q 2 R 0 ␦ M ␧ 3 R 0 q L y L q 2 R 0 exp w yc n x Ž . Ž 1 . 0 0 0 0 1 F

n M

Ž 1 q ␧ 1 . exp w yc0 ␤ 0 n x q 2 ␦ Ž M .

q exp yc1 n q

n

ž /Ž M

1 q ␧ 1 . log

␧ 3 Ž1y ␧ 1 . nrM

␤0 n

ž / M␦

.

To recapitulate, we have come thus far by assuming that M, n and nrM are large enough. By taking M G C log n for a large enough constant C, we can bound the last three terms above by expw yc2 nŽlog n.y1 x for a constant c 2 ) 0. Finally, ⺠ Ž TK Ž w nx x , w ny x . G n␥ K Ž x , y . q n ␧ . R0

F

Ý BŽ1. , . . . , BŽ R 0 .

⺠

ž


F 5 R 0 exp yc2 n Ž log n . F exp n Ž log n .

y1

y1

Ž Ž 1 q ␧ 1 . Ž log 5. rC y c2 .

F exp yc3 n Ž log n .

y1

/


400

for yet another constant c 3 ) 0, by increasing C further if necessary. This completes the proof of Lemma 7.1. One last word about the stage of the proof where we gave up the genuine exponential bound expw yCn x : the culprit is the factor Ý w g G Ž Rw0 . in Ž7.22., which we bounded above by Ž L1 y L 0 q 2. R 0 F Ž ␤ 0 nrM␦ .Ž1q ␧ 1 . n r M . To control this term, we need M to grow sufficiently fast with n. Then we can no longer assert that w 2 ␦ Ž M . ␧ 3 x Ž1y ␧ 1 . n r M decays exponentially, because we do not know the true decay rate of ␦ Ž M .. I 8. Proofs of the strong laws. In this section we prove the strong law part of Theorem 1 for K-exclusion, and Theorem 2 for the marching soldiers model. In this section the rates are constant ␣ Ž j . ' 1. 8.1. Proof of the strong law of Theorem 1. According to assumption Ž2.5., the initial configurations Ž␩nŽ i, 0.. are now defined on some common probability space Ž ⍀, F , P .. Augment this probability space to include the Poisson processes required for the graphical construction. Define the initial server configurations Ž z nŽ i, 0.. by Ž6.1.. Then construct the server processes z nŽ⭈. and w z n Ž i, 0. Ž⭈. and the interface processes ␰ z n Ž i, 0. Ž⭈. on Ž ⍀, F , P . according to the recipes of Section 4. As a corollary of our estimates we get the following lemma. LEMMA 8.1. t ) 0,

Suppose a0 s ␣ Ž j . s 1 for all j g Z. For any b, y g R and

Ž 8.1.

lim

1 n

nª⬁

␰ z n Žw n b x , 0. Ž w ny x , nt . s tg K Ž yrt .

P-a.s.

PROOF. By an argument similar to the one of Corollary 5.1, Lemmas 5.4 and 7.1 apply to deviations of ␰ as well as of TK and imply that ⬁

Ý ⺠ Ž ␰ Ž w ny x , nt . y tg K Ž yrt .

Ž 8.2.

G␧. -⬁

ns1

for any ␧ ) 0. Next note that the P-distribution of ␰ z n Žw n b x, 0. Žw ny x , nt . and the ⺠-distribution of ␰ Žw ny x , nt . are equal. Apply the Borel᎐Cantelli lemma. I By the reasoning that led to Ž6.3. under assumption Ž3.2., under the stronger assumption Ž2.5. we get that

Ž 8.3.

lim nª⬁

1 n

z n Ž w ny x , 0 . s U0 Ž y . ,

P-a.s. for all y g R.

Fix Ž x, t .. As argued in conjunction with Ž6.12. above, equations Ž2.8., Ž6.5., Ž8.1. and Ž8.3. imply that

Ž 8.4.

lim inf nª⬁

1 n

z n Ž w nx x , nt . G U Ž x , t . ,

P-a.s.

401


On the other hand, by Lemma 6.1 and Borel᎐Cantelli, lim sup nª⬁

1 n

z n Ž w nx x , nt . s lim sup nª⬁

1 n

␨n Ž r 1 , r 2 . ,

P-a.s.,

while by the development in Ž6.14., lim sup nª⬁

1 n

␨n Ž r 1 , r 2 . F U Ž x , t . q ␧r2,

P-a.s.

for any ␧ ) 0. In conclusion, we get

Ž 8.5.

lim nª⬁

1 n

z n Ž w nx x , nt . s U Ž x , t . ,

P-a.s.

and this suffices for the strong law of Theorem 1. 8.2. Proof of Theorem 2. Given the initial soldier locations ␴nŽ i, 0., define initial server configurations by z n Ž i , 0 . s ␴n Ž 0, 0 . y ␴n Ž yi , 0 . q iL1 ,

Ž 8.6.

i g Z.

This defines admissible server configurations with K s L1 q L2 . Next, construct the server processes z nŽ⭈. as in Section 4. Define the soldiers process ␴nŽ⭈. by

␴n Ž i , t . s ␴n Ž 0, 0 . y z n Ž yi , t . y iL1 ,

Ž 8.7.

i g Z.

The reader can check that this produces a process that satisfies the rules laid down for the marching soldiers in Section 2. The jump rule of z nŽ⭈. with K s L1 q L2 implies that ␴nŽ i, t . jumps right with rate 1 if ␴nŽ i, t . F ␴nŽ i y 1, t . q L2 y 1 and ␴nŽ i, t . F ␴nŽ i q 1, t . q L1 y 1. Assumption Ž2.22. and definition Ž8.6. imply that

Ž 8.8.

lim nª⬁

1 n

z n Ž w ny x , 0 . s U0 Ž y . ' V0 Ž 0 . y V0 Ž yy . q L1 y a.s.

for all y g R. From this assumption, we have just proved in Ž8.5. that lim

Ž 8.9.

nª⬁

1 n

z n Ž w nx x , nt .

s U Ž x , t . ' sup U0 Ž y . y tg L 1qL 2Ž Ž x y y . rt . 4

a.s.

y

for all x g R and t ) 0. The limit Ž8.9. and definition Ž8.7. then imply the limit of ny1␴nŽw nx x , nt .. Theorem 2 is thereby proved. Statement Ž2.24. follows from Ž2.20. and Theorem 2 applied to x s 0. 9. Properties of g K and f K . Suppose that the distribution of Ž z Ž i, 0.: i g Z. is such that z Ž0, 0. s 0 with probability 1 and that Ž␩ Ž i, 0. s z Ž i, 0. y z Ž i y 1, 0.: i g Z. is an ergodic process with expectation Ew ␩ Ž i, 0.x s ␳ . Then, with z nŽ i, 0. s z Ž i, 0. for all n, Ž6.3. is satisfied with U0 Ž y . s ␳ y. By Ž6.4., in


402

P ␣-probability, lim tª⬁

1 t

z Ž 0, t . s U Ž 0, 1 .

Ž 9.1.

s sup ␳ y y g K Ž yy . 4 ygR

s yf K Ž ␳ . , Ž . Ž . where we used 2.8 and 2.10 . On the other hand, recall that z Ž0, t . jumps leftward with rate ␣ Ž0. as long as ␩ Ž0, t . s z Ž0, t . y z Žy1, t . G 1 and ␩ Ž1, t . s z Ž1, t . y z Ž0, t . F K y 1. Then standard generator considerations permit us to write

Ž 9.2.

z Ž 0, t . s y␣ Ž 0 .

t

H0 I ␩ Ž 0, s . G 1, ␩ Ž 1, s . F K y 14 ds q M , t

where Mt is a martingale. Taking expectations gives

Ž 9.3.

E ␣ z Ž 0, t . s y␣ Ž 0 .

t

H0 P

␣

Ž ␩ Ž 0, s . G 1, ␩ Ž 1, s . F K y 1 . ds.

Clearly < z Ž0, t .< is bounded above by the total number of jump attempts in time w 0, t x which is PoissonŽ ␣ Ž0. t .-distributed. This gives uniform integrability, and, consequently, the limit in Ž9.1. holds also in expectation. So Ž2.14. and Ž3.5. are proved. Next we have a lemma about g K that will give us the properties of f K announced after Theorem 1. Recall that the limits of Proposition 5.1 and Corollary 5.1 required only ergodicity of Ž ␣ Ž j ... LEMMA 9.1.

Assume Ž ␣ Ž j .. is ergodic. We have these properties:

Ži. For y G 0, g K Ž y . G g Ky1Ž y . and for y - 0, g K Ž y . G g Ky1Ž y . y y. Žii. Suppose in addition that Ž ␣ Ž j .: j g Z. is reversible, so that Ž ␣ Ž j .: j g Z. sd Ž ␣ Žyj .: j g Z.. Then for all y g R, g K Žyy . s g K Ž y . q Ky. PROOF. For Ži., use the bijection p: LK ª LKy1 defined by

Ž 9.4.

pŽ i , j. s

½

Ž i, j. , Ž i q j, j . ,

j G 0, j - 0.

The image pŽ r . of a K-admissible path r may fail to be Ž K y 1.-admissible, but it is contained in a Ž K y 1.-admissible path. This is because, under p, Ž1, 0.-steps are preserved; a Ž K, y1.-step becomes either a single Ž K y 1, y1.-step or a Ž1, 0.-step followed by a Ž K y 1, y1.-step, and a Ž0, 1.-step becomes either a single Ž0, 1.-step or a Ž1, 0.-step followed by a Ž0, 1.-step. Since passage times are monotone under inclusion of paths, we get the inequality TK Ž i, j . F TKy1Ž pŽ i, j ... Passing to the limit Ž5.9. then gives

Ž 9.5.

␥K Ž x , y . F

This implies statement Ži..

½

␥ Ky 1 Ž x , y . ,

y G 0,

␥ Ky 1 Ž x q y, y . ,

y - 0.

403


For Žii. we use the bijection p: LK ª LK defined by pŽ i, j . s Ž i q Kj, yj .. Check that p is its own inverse, and that p preserves K-admissibility of lattice paths, by interchanging Ž K, y1.-steps with Ž0, 1.-steps, and by leaving Ž1, 0.-steps intact. The reversibility assumption implies that the distribution of the sitewise passage times is invariant under p: Ž␶ Ž i, j .: Ž i, j . g LK . sd Ž␶ Ž pŽ i, j ..: Ž i, j . g LK . where ␶ Ž i, j . s ␣ Ž j .y1␭Ž i, j . as in Section 5. From all this follows that TK Ž i, j . sd TK Ž pŽ i, j .., and consequently ␥ K Ž x, y . s ␥ K Ž x q Ky, yy .. This implies statement Žii.. I Lemma 9.1Žii. implies that any tangent to g K at x ) 0 must have slope in w yKr2, 0x . This and Ž2.10. imply that f K is nondecreasing on w 0, Kr2x . The symmetry Ž2.16. of f K follows from Ž2.10. and Lemma 9.1Žii., while the K-monotonicity Ž2.17. comes from Ž2.10. and Lemma 9.1Ži.. To get explicitly computable bounds, we restrict ourselves to the basic case without disorder. First we explain Remark 5.2. Definition Ž2.10., the convexity and continuity of g K and basic convex duality w Rockafellar Ž1970.x imply that g K Ž y . s sup

Ž 9.6.

0F ␳ FK

fK Ž ␳ . y x ␳ 4 .

For K s 1 we deduce, from Ž2.15. and Bernoulli equilibria ␯␳ , the well-known flux f 1Ž ␳ . s ␳ Ž1 y ␳ ., and then Ž9.6. gives

¡yy, ¢0,

g Ž y . s~ Ž 1r4 . Ž 1 y y . 1

y F y1, 2

,

y1 F y F 1, y G 1.

From homogeneity Ž5.18. and the identity ␥ 1Ž g 1Ž y ., y . s 1 for y g w y1, 1x , one deduces ␥ 1Ž x, y . s Ž'x q x q y . 2 . Equations Ž5.23. Žwith a1 s 1. and Ž9.5. then imply the boundary values Ž5.24.. Lemma 9.1Ži. gives a lower bound for g K in terms of g 1. To get an upper bound, we utilize Rost’s Ž1981. calculation Ž5.14.. Define a convex function h K by

'

¡yKy, Ž 9.7.

~

y F y1,

Ž 1 y 'y y .

2

y Ky,

h K Ž y . s Kr Ž K q 2 . y Kyr2,

Ž 1 y 'y .

¢0, LEMMA 9.2.

2

,

2

y1 F y F y4r Ž K q 2 . , 2

2

y4r Ž K q 2 . F y F 4r Ž K q 2 . , 2

4r Ž K q 2 . F y F 1, y G 1.

Suppose ␣ Ž j . s 1 for all j. Then g K Ž y . F h K Ž y . for all y g R.


404

PROOF. Let Ž i, j . g N 2 . In the definition Ž5.12. of the last-passage time V Ž i, j ., set v Ž i, j . s ␭Ž i, j .. Since the up-right paths r ⬘ in Ž5.12. are K-admissible, TK Ž i, j . G V Ž i, j .. By passing to the limits Ž5.9. and Ž5.14., we get ␥ K Ž x, y . G Ž'x q y . 2 for x, y ) 0, and consequently g K Ž y . F Ž1 y y . 2 for 0 F y F 1. An application of Lemma 9.1Žii. then gives

'

'

¡yKy,

Ž 9.8.

˜K Ž y . s gK Ž y . F h

y F y1,

~ Ž1 y 'y y . 2

1y y. ,

¢0,Ž

'

2

y Ky,

y1 F y F 0, 0 F y F 1, 1 F y.

˜ K is not convex while g K is convex, so we can replace h ˜ K in The function h the inequality by its convex hull h K , the greatest convex function majorized ˜K . I by h Substituting the upper bound h K for g K in Ž2.10. gives the upper bound Ž2.19. for f K . The function on the right-hand side of Ž2.19. has a corner at ␳ s Kr2. This corresponds to the linear segment of slope yKr2 that h K has on an interval around 0. 10. A non-Markovian K-exclusion. Let ␤ Ž l, i ., Ž l, i . g Z 2 , be Ž0, ⬁.valued i.i.d. random variables with common cumulative distribution function F. In this section we study a non-Markovian totally asymmetric K-exclusion that uses the process Di s ␤ Ž l, i .: l g Z4 as the waiting times for jumps from site i to i q 1. These are the informally stated rules: Let t 0 be the last time a particle jumped from site i to i q 1, or let t 0 s 0 if we are just starting the process. Let t 1 be the first time in w t 0 , ⬁. such that

Ž 10.1.

␩ Ž i , t 1 . G 1 and ␩ Ž i q 1, t 1 . F K y 1.

Let ␤ Ž l, i . be the next unused waiting time from Di . Then at time t 2 s t 1 q ␤ Ž l, i . one particle jumps from site i to i q 1; in other words, ␩ Ž i, t 2 . s ␩ Ž i, t 2 y . y 1 and ␩ Ž i q 1, t 2 . s ␩ Ž i q 1, t 2 y . q 1. Notice that the only events involving sites i and i q 1 that can happen in the time interval w t 1 , t 2 . are particles jumping from i q 1 to i q 2, or from i y 1 to i. Neither event violates the precondition Ž10.1. for the jump from i to i q 1, so this jump can be legitimately executed at time t 2 . The process Di need not be bi-infinite for this description, but later this property will be useful. The jumping rule can be summarized like this: as soon as Ž10.1. holds, pick an F-distributed waiting time ␤ independently of everything else, and let one particle jump from i to i q 1 after time ␤ . The possibility of simultaneous jumps is not a problem, so we make no continuity assumptions on F. We shall not attempt a more technical description of the process at this point. In Section 10.1 below we define the process through a rigorous construction of the corresponding server process z Ž⭈., as was done in Section 4 for the Markovian K-exclusion.

405


Otherwise the basic set-up is the same as in Section 2. We assume a sequence ␩nŽ⭈. of processes that satisfies the weak law Ž2.4. at time zero. Let us normalize the waiting times by assuming Ew ␤ Ž l, i .x s 1 so that Ž2.6. becomes part of the conclusion again. ŽThe case Ew ␤ Ž l, i .x s ⬁ gives trivial hydrodynamic behavior where nothing moves, for the reason outlined in the last paragraph of Section 6.. THEOREM 4. There exists a nonnegative convex function g K that depends on F and has property Ž2.6., and such that the following statement holds: if assumption Ž2.4. is satisfied and uŽ x, t . is defined by Ž2.7. ᎐ Ž2.9., then for each t ) 0, ␾ g C0 ŽR., and ␧ ) 0,

Ž 10.2.

lim Pn ␲n Ž nt , ␾ . y

nª⬁

ž


G ␧ s 0.

/

If we wanted a strong law, we would need some boundedness assumptions on F to make Lemma 7.1 valid. To prove Theorem 4 we first construct a process ␩ Ž⭈. that operates according to the rules laid down above, and then we observe how the arguments of Sections 5 and 6 work in this more general setting. 10.1. Construction of the process. As in Section 4, we construct the server process z Ž⭈. whose particles jump to the left and then define the customer process ␩ Ž⭈. by Ž4.1.. Assume we are given an initial configuration Ž z Ž i, 0.: i g Z. such that Ž4.1. holds for t s 0, where Ž␩ Ž i, 0.: i g Z. is the desired initial K-exclusion configuration. In particular, the inequalities

Ž 10.3.

0 F z Ž i q 1, t . y z Ž i , t . F K

hold at t s 0. The informal rule for the evolution of z Ž⭈. follows from the earlier description for ␩ Ž⭈., and now we specify that ␤ Ž l, i . is the waiting time for server z Ž i . to execute the jump from site l q 1 to l. If z Ž i, 0. ) l, let t 1 be the earliest moment at which these three conditions hold:

Ž 10.4. z Ž i y 1, t1 . F l, z Ž i , t1 . s l q 1 and

z Ž i q 1, t 1 . F l q K .

Then at time t 2 s t 1 q ␤ Ž l, i . server z Ž i . jumps to site l. We shall construct the trajectories z Ž i, t . by rigorously defining the jump times. Let ␶ Ž l, i . denote the earliest time t such that z Ž i, t . F l holds. There is one technical twist; a particular z Ž i . may never reach certain sites l because z Ž i . comes to a permanent halt during the dynamics. Then the corresponding ␶ Ž l, i .’s must be infinite. This happens if in the initial K-exclusion configuration either ␩ Ž i, 0. s 0 for all sites i far enough to the left or ␩ Ž i, 0. s K for all sites i far enough to the right. For the initial z Ž i, 0.’s, the corresponding conditions are these:

Ž 10.5.

l# s lim z Ž i , 0 . ) y⬁ iªy⬁


406

or

Ž 10.6. there exists i* - ⬁ such that z Ž i q 1, 0 . s z Ž i , 0 . q K for i G i*. In case Ž10.5., no z Ž i . ever makes it past l#, while in case Ž10.6., z Ž i . never makes it beyond z Ž i*, 0. q K Ž i y i*., for all i. This leads us to define the domain I of ␶ Ž l, i . as follows: I s Z2

if neither Ž 10.5. nor Ž 10.6. holds,

I s Ž l, i . g Z 2 : l G l# 4

if Ž 10.5. holds but not Ž 10.6. ,

I s Ž l, i . g Z : l G z Ž i*, 0 . q K Ž i y i* . 4 2

Ž 10.7.

if Ž 10.6. holds but not Ž 10.5. and I s Ž l, i . g Z 2 : l G max l#, z Ž i*, 0 . q K Ž i y i* .

4

if both Ž 10.5. and Ž 10.6. hold. Outside of this domain we set

Ž 10.8.

␶ Ž l, i . s ⬁ for Ž l, i . f I .

Let I0 s Ž l, i .: l G z Ž i, 0.4 : I . The initial configuration Ž z Ž i, 0.: i g Z. dictates that

Ž 10.9.

␶ Ž l, i . s 0 for Ž l, i . g I0 .

The remaining values are determined by the equation

Ž 10.10.

␶ Ž l, i . s ␤ Ž l, i . q max ␶ Ž l, i y 1 . , ␶ Ž l q 1, i . , ␶ Ž l q K , i q 1 . 4 , Ž l, i . g I _ I0 .

Equation Ž10.10. incorporates the dynamics, starting with the ‘‘initial values’’ Ž10.9.. It is the formalization of the description around Ž10.4. above. Observe that ‘‘time increases leftward’’ on the lattice Ž l, i . in the sense that ␶ Ž l, i . increases as l decreases, because z Ž i . jumps leftward. We need to prove that Ž10.8. ᎐ Ž10.10. define all ␶ Ž l, i .’s, and thereby the dynamics z Ž⭈.. Notice that Ž10.8. and Ž10.10. are consistent in this sense: it follows from the definition Ž10.7. of I that if Ž l, i y 1., Ž l q 1, i . and Ž l q K, i q 1. all lie in I , then so does Ž l, i .. In other words, equation Ž10.10. cannot define ␶ beyond the domain I . Furthermore, since ␤ Ž l, i . ) 0, Ž10.9. and Ž10.10. together imply that ␶ Ž l, i . s 0 iff Ž l, i . g I0 . To prevent the jump times from accumulating at a finite value, assume that l

Ž 10.11.

Ý

␤ Ž m, i . s ⬁ for all l, i.

msy⬁

By the strong law of large numbers for i.i.d. random variables, a.e. realization ␤ Ž l, i .4 satisfies Ž10.11..


407

LEMMA 10.1. For given ␤ Ž l, i .: Ž l, i . g Z 2 4 that satisfy Ž10.11., there is a unique function ␶ on Z 2 that satisfies equations Ž10.8. ᎐ Ž10.10.. Furthermore, this function ␶ satisfies lim l ªy⬁ ␶ Ž l, i . s ⬁ for all i. PROOF. We first show that Ž10.9. and ␤ Ž l, i .4 determine ␶ Ž l, i . for each Ž l,i . g I _ I0 through Ž10.10.. To do this by induction, let In be the set of Ž l, i . g I such that

Ž l, i . q Ž 0, yn1 . q Ž n 2 , 0 . q Ž Kn 3 , n 3 . g I0 for all choices of n1 , n 2 , n 3 G 0 such that n1 q n 2 q n 3 s n. First note that I0 : I1 : I2 : ⭈⭈⭈ because by Ž10.3., I0 is closed under the maps

Ž 10.12.

Ž l, i . ¬ Ž l, i . q Ž 0, yn1 . q Ž n 2 , 0 . q Ž Kn 3 , n 3 .

for all n1 , n 2 , n 3 G 0. Second, Ž10.10. defines ␶ Ž l, i . inductively over successive In’s. Assume that ␶ Ž l, i . is defined for all Ž l, i . g In . If Ž l, i . g Inq1 , then Ž l, i y 1., Ž l q 1, i . and Ž l q K, i q 1. all lie in In , and equation Ž10.10. defines ␶ Ž l, i .. Next we argue that I s D n G 0 In . Pick and fix Ž l, i . g I . By checking the different cases in Ž10.7., one sees that it is possible to choose m large enough so that Ž l, i y m., Ž l q m, i ., Ž l q Km, i q m. all lie in I0 . It follows that Ž l, i . g I3 m , because n1 q n 2 q n 3 s 3m forces some n i G m, and because I0 itself is closed under the maps Ž10.12.. We have showed that ␶ is defined on all of Z 2 . Uniqueness is essentially contained in the previous paragraphs. Suppose ˜ ␶ is a different function that satisfies Ž10.8. ᎐ Ž10.10.. Then ␶ Ž l, i . / ˜ ␶ Ž l, i . for some Ž l, i . g I _ I0 . Then by Ž10.10., ␶ and ˜ ␶ must differ at one of Ž l, i y 1., Ž l q 1, i ., or Ž l q K, i q 1.. Proceeding inductively, we eventually conclude that ␶ and ˜ ␶ must differ at some Ž l, i . g I0 , contradicting Ž10.9.. Fix i. If Ž l, i . f I for some l, then lim l ªy⬁ ␶ Ž l, i . s ⬁ is immediate. Suppose Ž l, i . g I for all l. Set l 0 s z Ž i, 0. y 1. Since Ž l 0 , i . g I _ I0 , we can apply Ž10.10. repeatedly to get, for any l - l 0 ,

␶ Ž l, i . G ␤ Ž l, i . q ␶ Ž l q 1, i . G ␤ Ž l, i . q ␤ Ž l q 1, i . q ␶ Ž l q 2, i . l0

G ⭈⭈⭈ G

Ý ␤ Ž m, i . . msl

Now lim l ªy⬁ ␶ Ž l, i . s ⬁ follows from Ž10.11.. I Having defined the jump times ␶ Ž l, i ., we define the dynamics by

Ž 10.13.

z Ž i , t . s l for ␶ Ž l, i . F t - ␶ Ž l y 1, i .

for all l, i g Z.

Lemma 10.1 guarantees that z Ž i, t . is defined for all time 0 F t - ⬁. Equation Ž10.10. implies that Ž10.3. holds for all t ) 0.


408

10.2. The auxiliary processes. As in Section 4, given an initial configuration Ž z Ž i, 0.: i g Z. denote by w zŽ i, 0. Ž⭈. the server process with the special initial configuration

Ž 10.14.

w zŽ i , 0. Ž j, 0 . s

½

z Ž i , 0. , z Ž i , 0 . q Kj,

j G 0, j - 0.

Our goal is to prove a result that corresponds to the central Lemma 4.2. First we define further requirements satisfied by a ‘‘good’’ realization of ␤ Ž l, i .4 . k

Ž 10.15.

Ý

␤ Ž l, j . s ⬁ for all l, k

jsy⬁

and ⬁

Ž 10.16.

Ý ␤ Ž l q Ž j y k. K , j. s ⬁

for all l, k.

jsk

A.e. realization ␤ Ž l, i .4 satisfies Ž10.11., Ž10.15. and Ž10.16.. Corresponding to Ž4.4., we stipulate that Particle w zŽ i, 0. Ž j . uses waiting time ␤ Ž l, i q j . for its jump

Ž 10.17. from site l q 1 to l.

The point of Ž10.17. is that both z Ž k . and w zŽ i, 0. Ž k y i . use the same waiting time ␤ Ž l, k . to jump from site l q 1 to l. LEMMA 10.2. Assume we are given an initial configuration Ž z Ž i, 0.: i g Z. for the z Ž⭈.-process and a realization ␤ Ž l, i .4 of the waiting times that satisfies Ž10.11., Ž10.15. and Ž10.16.. Construct the processes w zŽ i, 0. Ž⭈. according to the description of Section 10.1 and rule Ž10.17.. Then the following equality holds for all k g Z and t G 0:

Ž 10.18.

z Ž k , t . s sup w zŽ i , 0. Ž k y i , t . . igZ

PROOF. Let ␴ i Ž l, j . be the earliest time t when w zŽ i, 0. Ž j, t . F l. By the construction Ž10.8. ᎐ Ž10.10. and the initial condition Ž10.14., the domain of ␴ i is Z 2 for each i,

Ž 10.19.

␴ i Ž l, j . s 0 for l G z Ž i , 0 . q K ⭈ min j, 0 4

and

␴ i Ž l, j . s ␤ Ž l, i q j .

Ž 10.20.

q max ␴ i Ž l, j y 1 . , ␴ i Ž l q 1, j . , ␴ i Ž l q K , j q 1 . 4 for l - z Ž i , 0 . q K ⭈ min j, 0 4 .

Let I and I0 be as defined by Ž10.7. and by the sentence after Ž10.8., on the basis of the given Ž z Ž i, 0.: i g Z.. Set

Ž 10.21.

˜␶ Ž l, k . s sup ␴ i Ž l, k y i . . igZ

409


Our first goal is to show that ˜ ␶ satisfies Ž10.8. ᎐ Ž10.10., and therefore by Lemma 10.1, ˜ ␶ s ␶. Condition Ž10.9. for ˜ ␶ follows immediately because for all i, ␴ i Ž l, k y i . s 0 for l G z Ž k, 0. by Ž10.19. and Ž10.3.. To deduce Ž10.10. for ˜ ␶ , note first that

˜␶ Ž l, k . s max 0, sup ␴ i Ž l, k y i . ,

½

Ž 10.22.

5

igJ l, k

where

Ž 10.23.

Jl , k s i g Z: z Ž i , 0 . q K ⭈ min k y i , 0 4 ) l 4 .

Equation Ž10.22. is true because ␴ i Ž l, k y i . s 0 for i f Jl, k , by Ž10.19.. By Ž10.3., Jl, k s ⭋ when l G z Ž k, 0., so we need the comparison with zero on the right-hand side of Ž10.22.. ŽBy convention, the supremum of an empty set is y⬁.. If l - z Ž k, 0. w equivalently, Ž l, k . f I0 x , then k g Jl, k , so Jl, k / ⭋. Then the comparison with zero is not needed in Ž10.22. because ␴ i G 0 always. Now we can deduce Ž10.10. for Ž l, k . g I _ I0 . First,

˜␶ Ž l, k . s sup ␴ i Ž l, k y i . igJ l, k

s sup

␤ Ž l, k . q max ␴ i Ž l, k y i y 1 . , ␴ i Ž l q 1, k y i . ,

igJ l, k

␴ i Ž l q K , k y i q 1. 4 by Ž 10.20.

Ž 10.24. s ␤ Ž l, k . q max

½ sup ␴ Ž l, k y i y 1. , sup ␴ Ž l q 1, k y i . , i

i

igJ l, k

igJ l , k

sup ␴ i Ž l q K , k y i q 1 . igJ l, k

5

F ␤ Ž l, k . q max ␶˜ Ž l, k y 1 . , ␶˜ Ž l q 1, k . , ˜ ␶ Ž l q K , k q 1. 4 , where the last inequality follows from Ž10.21.. On the other hand, one can check that Jl, k = Jl, ky1 j Jlq1, k j JlqK , kq1. Then we can continue from the next-to-last line in Ž10.24. by

˜␶ Ž l, k . G ␤ Ž l, k . q max 0,

½

Ž 10.25.

sup ␴ i Ž l, k y i y 1 . , sup ␴ i Ž l q 1, k y i . , igJ l, ky1

igJ lq1 , k

sup

␴ i Ž l q K , k y i q 1.

igJ lqK , kq1

5

s ␤ Ž l, k . q max ˜ ␶ Ž l, k y 1 . , ˜ ␶ Ž l q 1, k . , ˜ ␶ Ž l q K , k q 1. 4 , where the last step now comes from Ž10.22.. We have verified that ˜ ␶ satisfies Ž10.10..


410

It remains to check Ž10.8., namely, that ˜ ␶ Ž l, k . s ⬁ for Ž l, k . f I . Two cases, according to Ž10.5. and Ž10.6., need to be considered. CASE I. Suppose l# ) y⬁ and l - l#. Let i be a large negative integer, large enough so that z Ž i, 0. s l# and k y i 4 0. By repeated application of Ž10.20.,

␴ i Ž l, k y i . G ␤ Ž l, k . q ␴ i Ž l, k y i y 1 . G ␤ Ž l, k . q ␤ Ž l, k y 1 . q ␴ i Ž l, k y i y 2 .

Ž 10.26.

k

G ⭈⭈⭈ G

Ý

␤ Ž l, j . q ␴ i Ž l, k y i y i 0 .

jskyi 0q1

as long as k y i y i 0 G 0 because then Ž10.20. is valid. Take i 0 s k y i and combine with Ž10.21. to write k

˜␶ Ž l, k . G ␴ i Ž l, k y i . G Ý ␤ Ž l, j . . jsiq1

Now ˜ ␶ Ž l, k . s ⬁ follows from Ž10.15. by letting i o y⬁. CASE II. Suppose i* - ⬁ and l - z Ž i*, 0. q K Ž k y i*.. Let i 4 max k, i*4 , and i 0 F i y k. The assumption on Ž l, k . and z Ž i, 0. s z Ž i*, 0. q K Ž i y i*. imply that l q i 0 K - z Ž i, 0. q K Ž k y i q i 0 ., and we can again apply Ž10.20. inductively to write

␴ i Ž l, k y i . G ␤ Ž l, k . q ␴ i Ž l q K , k y i q 1 .

Ž 10.27.

G ␤ Ž l, k . q ␤ Ž l q K , k q 1 . q ␴ i Ž l q 2 K , k y i q 2 . kqi 0y1

G ⭈⭈⭈ G

Ý

␤ Ž l q Ž j y k . K , j . q ␴ i Ž l q i0 K , k y i q i0 . .

jsk

Take i 0 s i y k, use Ž10.21., let i p ⬁ and apply Ž10.16. to conclude that

˜␶ Ž l, k . s ⬁.

This concludes the first step of the proof of Lemma 10.2. Since ˜ ␶ satisfies Ž10.8. ᎐ Ž10.10., Lemma 10.1 implies that

Ž 10.28.

␶ Ž l, k . s sup ␴ i Ž l, k y i . . igZ

This implies that ‘‘G ’’ holds in Ž10.18., because no w zŽ i, 0. Ž k y i . arrives in l any later than z Ž k .. It remains to argue that some w zŽ i, 0. Ž k y i . remains at site l q 1 with z Ž k . until time ␶ Ž l, k .. For the dynamics of z Ž⭈., we only concern ourselves with Ž l, k . g I in Ž10.28.. Fix such a pair Ž l, k .. The next step is to argue that the supremum in Ž10.28. is always realized at some finite i. To this end we show that ␴ i Ž l, k y i . s 0 for all large enough positive and negative i. By Ž10.19., this follows from showing that

Ž 10.29.

l G z Ž i , 0 . q K ⭈ min k y i , 0 4 .


411

First for large negative i, simply pick i 0 - k so that z Ž i 0 , 0. F l, and then Ž10.29. holds for i F i 0 . For large positive i we need two cases. Suppose first that Ž10.6. does not hold. Let u i denote the number of indices j g k q 1, . . . , i4 such that z Ž i, 0. y z Ž i y 1, 0. F K y 1. Since Ž10.6. does not hold, u i p ⬁ as i p ⬁. Pick i 1 ) k large enough so that u i 1 G z Ž k, 0. y l. Then for i G i 1 , z Ž i , 0. F z Ž k , 0. q Ž i y k y u i . K q u i Ž K y 1. s z Ž k , 0. y u i q K Ž i y k . F l q K Ž i y k. , and Ž10.29. follows. On the other hand, if Ž10.6. holds, then Ž l, k . g I forces l G z Ž i*, 0. q K Ž k y i*.. This, together with z Ž i, 0. s z Ž i*, 0. q K Ž i y i*. for i G i*, implies Ž10.29.. To summarize, for each Ž l, k . g I , there exists an index i 0 such that

Ž 10.30.

␶ Ž l, k . s sup ␴ i Ž l, k y i . s ␴ i 0 Ž l, k y i 0 . . igZ

We claim that for Ž l, k . g I ,

Ž 10.31.

w zŽ i 0 , 0. Ž k y i 0 , t . s z Ž k , t . s l q 1 for ␶ Ž l q 1, k . F t - ␶ Ž l, k . .

As already observed, w zŽ i 0 , 0. Ž k y i 0 , t . F z Ž k , t . s l q 1 by virtue of Ž10.28.. On the other hand, ␴ i 0 Ž l, k y i 0 . is the earliest time for w zŽ i 0 , 0. Ž k y i 0 , t . F l to hold, so w zŽ i 0 , 0. Ž k y i 0 , t . G l q 1 for t - ␶ Ž l, k .. This proves Ž10.31.. Consequently, Ž10.18. holds for t - ␶ Ž l, k . whenever ␶ Ž l, k . - ⬁. It remains to consider the case ␶ Ž l q 1, k . F t - ⬁ s ␶ Ž l, k .. Now z Ž k, t . has come to a halt at site l q 1, and there are two possible causes for this. The first possibility is that z Ž k, t . s l q 1 s l# ) y⬁. By the calculation in Ž10.26., we can choose i < k so that z Ž i, 0. s l# and ␴ i Ž l, k y i . ) t. Then w zŽ i, 0. Ž k y i, t . s l q 1 s z Ž k, t ., and Ž10.18. is valid again. The other possibility is that Ž10.6. holds and z Ž k, t . s z Ž i*, 0. y K Ž i* y k .. Then we choose i 4 max i*, k 4 large enough so that ␴ i Ž l, k y i . ) t, as can be done by the calculation in Ž10.27.. Then w zŽ i, 0. Ž k y i, t . s l q 1 s z Ž k, t .. This concludes the proof of Ž10.18.. I 10.3. The growth model. The growth model is formulated exactly as in Section 5. Now ␣ Ž j . s 1 for all j, and the Ž ␭Ž i, j .: Ž i, j . g Z 2 . are i.i.d. F-distributed instead of exponentially distributed. The passage times TK Ž i, j . are defined in terms of K-admissible paths as in Ž5.5., and the interface process ␰ Ž j, t . is defined by Ž5.8..


412

PROPOSITION 10.1. Let the common distribution F of the i.i.d. ␤ Ž l, i .4 be any distribution on Ž0, ⬁. that satisfies Ew ␤ Ž l, i .x s 1. Then there is a concave function ␥ K defined on UK such that 1

T Ž w nx x , w ny x . s ␥ K Ž x , y . n K holds almost surely, for all Ž x, y . g UK . Either ␥ K s ⬁ on all of UK or ␥ K - ⬁ on all of UK . The equation

Ž 10.32.

lim

nª⬁

g K Ž y . s inf x ) 0: Ž x , y . g UK , ␥ K Ž x , y . G 1 4 defines a finite, convex, continuous, nonincreasing function on R with property Ž2.6.. For any y g R and t ) 0,

Ž 10.33.

lim nª⬁

1 n

␰ Ž w ny x , nt . s tg K Ž yrt .

a.s.

PROOF. The proof of Proposition 5.1 applies with some adjustments. The point to note is that a nonnegative superadditive process converges almost surely with or without the moment bound, as long as the other assumptions are met. This can be seen by applying a truncation argument to the basic subadditive ergodic theorem. The first step of the proof of Proposition 5.1 now defines a w 0, ⬁x -valued function ␥ K Ž k, l . on LK that gives the limit Ž10.32. for Ž x, y . s Ž k, l .. Suppose that ␥ K Ž i, j . s ⬁ for some Ž i, j . g LK . For any Ž k, l . g LK , find m large enough such that Ž mk, ml . g Ž i, j . q LK . Then m␥ K Ž k , l . s ␥ K Ž mk , ml . G ␥ K Ž i , j . . This shows that either ␥ K Ž k, l . - ⬁ for all Ž k, l . g LK or ␥ K Ž k, l . s ⬁ for all Ž k, l . g LK . In the former case, follow the proof of Proposition 5.1, in the latter case modify that proof to show that Ž10.32. holds with ␥ K Ž x, y . ' ⬁. The limit Ž10.33. follows as Corollary 5.1 did. Note that ␥ K s ⬁ poses no problem for the definition of g K . In this case, g K Ž y . s 0 for y G 0 and g K Ž y . s yKy for y F 0. I Another collection of interface processes is defined, as in Section 4, by

␰ zŽ i , 0. Ž j, t . s z Ž i , 0 . y w zŽ i , 0. Ž j, t . .

Ž 10.34.

LEMMA 10.3. For any sequence of possibly random initial configurations Ž z nŽ i, 0.: i g Z., any sequence of indices i n and any y g R and t ) 0,

Ž 10.35.

lim nª⬁

1 n

␰ z n Ž i n , 0. Ž w ny x , nt . s tg K

y

ž / t

in probability.

PROOF. Suppose first that we have a deterministic initial configuration Ž z Ž i, 0.: i g Z.. We shall show that for any fixed i, j and t,

Ž 10.36.

␰ zŽ i , 0. Ž j, t . sd ␰ Ž j, t . .


413

Then Ž10.35. follows from Ž10.33. for a deterministic sequence of initial configurations. Subsequently, Ž10.35. follows for random Ž z nŽ i, 0.: i g Z. by integrating over the distribution of z nŽ i n , 0.. Fix i. By Ž10.34., the time t when ␰ zŽ i, 0. Ž j, t . s l first holds is the same as iŽ Ž ␴ z i, 0. y l, j ., the time when server w zŽ i, 0. Ž j . first reaches site z Ž i, 0. y l. By Ž5.8., the time when ␰ Ž j, t . s l first holds is the past-passage time TK Ž l, j .. To prove Ž10.36. and thereby the lemma, we show the equality in distribution of the time processes

Ž 10.37.

␴ i Ž z Ž i , 0. y l, j . : Ž l, j . g LK 4 sd TK Ž l, j . : Ž l, j . g LK 4 .

To establish this, we choose the sitewise passage times ␶ Ž l, j . from a realization of the waiting times ␤ Ž l, i .. Set ␶ Ž l, j . s ␤ Ž z Ž i, 0. y l, i q j . for Ž l, j . g LK . By Ž5.5., for Ž l, j . g LK , TK Ž l, j .

Ž 10.38.

s ␶ Ž l, j . q max TK Ž l, j y 1 . , TK Ž l y 1, j . ,TK Ž l y K , j q 1 . 4 s ␤ Ž z Ž i , 0 . y l, i q j . q max TK Ž l, j y 1 . , TK Ž l y 1, j . , TK Ž l y K , j q 1 . 4 ,

with initial T Ž l, j . s 0 for Ž l, j . f LK . A comparison with Ž10.19. and Ž10.20. shows that ␴ i Ž z Ž i, 0. y l, j . satisfies the same recursion Ž10.38., with the same initial values ␴ i Ž z Ž i, 0. y l, j . s 0 for Ž l, j . f LK . It follows inductively that, with this particular choice of ␶ Ž l, j .’s, TK Ž l, j . actually equals ␴ i Ž z Ž i, 0. y l, j .. For arbitrary i.i.d. F-distributed passage times ␶ Ž l, j . we then have the equality in distribution Ž10.37.. I Now rewrite Ž10.18. as

Ž 10.39.

z Ž k , t . s sup z Ž i , 0 . y ␰ zŽ i , 0. Ž k y i , t . 4 . igZ

10.4. Proof of Theorem 4. The route from Ž10.39. to Ž10.2. is the same as the one taken in Section 6. The goal is again to prove Ž6.4., with ␣ ' 1. The easy half Ž6.12. is easy again and for the same reasons. The harder half required two lemmas: Lemma 6.1, which allows us to exclude the extreme values of i in Ž10.39., and Lemma 4.3 about preserving the ordering of particles. Both lemmas are valid again, and hence the reasoning Ž6.14. can be repeated to deduce Ž6.15., which together with Ž6.8. completes the proof of Ž10.2.. The Borel᎐Cantelli estimate Ž5.26. was required only for the disordered weak law, and we do not need it now, although it would be valid here too. With this we consider Theorem 4 proved. REFERENCES ALDOUS, D. and DIACONIS, P. Ž1995.. Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199᎐213.

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