Trap models with vanishing drift: Scaling limits and ageing regimes

arXiv:1003.1490v1 [math.PR] 7 Mar 2010

Trap models with vanishing drift: Scaling limits and ageing regimes Nina Gantert, Peter Mörters, Vitali Wachtel March 9, 2010 Abstract We discuss the long term behaviour of trap models on the integers with asymptotically vanishing drift, providing scaling limit theorems and ageing results. Depending on the tail behaviour of the traps and the strength of the drift, we identify three different regimes, one of which features a previously unobserved limit process.

1

Introduction

Trap models are a particularly simple class of stochastic processes in random environment, which have recently attracted a lot of attention. To describe the setup of most trap models, suppose a graph with finite degree is given. To each of the vertices v of the graph we associate an independent random variable τv chosen according to a suitable class of heavytailed distributions. Given this random environment, the trap model is a continuous-time nearest neighbour random walk on the graph such that the exponential holding time at a vertex v has a mean proportional to τv . Therefore vertices v with large values τv act as traps in which the random walk spends a larger amount of time than in vertices with small values of v. Different trap models arise by varying the underlying graph and the drift of the random walk. The main purpose of trap models is to serve as a phenomenological model describing how a physical system out of equilibrium moves in an energy landscape. Here vertices with large trap values represent energetically favourable states in which the system tends to remain for longer. Most results on trap models are about the phenomenon of ageing, which means that in such a system the time spans during which the system does not change its state are increasing as the system gets older. Trap models offer a simple explanation for ageing: Roughly speaking, the older the system, the more space it has explored, and therefore the deeper the trap it is stuck in. Let us also mention here some interesting papers exhibiting the ageing phenomenon in some other models, for example for spherical spin glasses (Ben Arous et al. (2001)), the random energy model with Glauber dynamics (Ben Arous et al. (2003)) and the parabolic Anderson model with heavy tailed potential (Mörters et al. (2009)).

1

Trap models were introduced into the physics literature by Bouchaud (1992) and interest in the mathematical community was created through the pioneering work of Fontes et al. (2002) and Ben Arous et al. (2006). An excellent survey over the mathematical literature ˇ y (2006). on trap models is provided in the lecture notes of Ben Arous and Cern´ Understanding the ageing phenomenon is closely linked to scaling limit theorems for the trap models. For driftless trap models on the lattice Zd it was shown that, on suitable path spaces, • if d = 1 the rescaled trap model converges to a singular diffusion without drift, which is often called the Fontes-Isopi-Newman diffusion, see Fontes et al. (2002); • if d ≥ 2 the rescaled trap model converges to the fractional-kinetics process, which is a self-similar non-Markovian process, obtained as the time change of a d-dimensional ˇ y Brownian motion by the inverse of an independent stable subordinator, see Ben Arous and Cern´ (2007). ˇ y (2009) identified the fractional-kinetics process as the More recently, Barlow and Cern´ scaling limit for a class of random walks with unbounded conductances and for the socalled non-symmetric trap models on Zd , d ≥ 3, which have a drift depending locally on the trap environment. In the present paper we focus on trap models on Z with a drift, which does not depend ˇ y (2006). In on the trap environment, addressing a question posed in Ben Arous and Cern´ our first main result, Theorem 2.1, we look at the scaling limits of trap models with an asymptotically vanishing drift, and identify three regimes: • In a regime where the drift vanishes slowly, the rescaled trap model converges to the inverse of a stable subordinator. Zindy (2009), using a different method of proof, identified the same process as the scaling limit for trap models with constant drift. • In a regime where the drift vanishes quickly the rescaled trap model converges to a Fontes-Isopi-Newman diffusion, the same process as in the driftless case. • In a critical intermediate regime the rescaled trap model converges to a singular diffusion with drift, which we call the Fontes-Isopi-Newman diffusion with drift. This process has not been identified as limit process in any other case before. Our second main result, stated as Theorems 2.2 and 2.4, refer to the ageing behaviour in trap models on Z with vanishing drift. To this end we study the asymptotics of the depth of the trap in which the particle is at any given time, or, in other words, the environment from the point of view of the particle. This allows us to identify an ageing exponent 0 < γ ≤ 1 such that the probability P Xt = Xt+s for all 0 ≤ s ≤ tγ as t ↑ ∞,

(averaged over the trap environment) converges to a value strictly between zero and one. Again there is a qualitatively different behaviour between the case of slowly vanishing drift on the one hand, and rapidly vanishing and critical drift on the other. Only in the 2

case of constant drift do we have an ageing exponent γ = 1, all regimes with vanishing drift (as t ↑ ∞) lead to sublinear ageing, i.e. exponents γ < 1. This is in marked contrast to the behaviour of the two-point function P{Xt = Xt+tγ } for which we expect a nontrivial limit when γ = 1 in all cases, a fact which is rigorously established in the driftless case in Fontes et al. (2002) and in the case of fixed drift in Zindy (2009). In the following section we give the precise formulation of our main results. We then proceed to prove our scaling limit theorems in the three regimes in Sections 3, 4 and 5, and the two regimes of ageing results in Section 6 and 7.

2

Statement of the main results

Fix 0 < α < 1 and let (τz : z ∈ Z) be an independent family of random variables with lim xα P {τz > x} = 1 .

x↑∞

(1)

Given this trap environment and jump probabilities p, q ∈ [0, 1], q = 1 − p, we define the Markov chain on Z with transition rates qi,i+1 = p τi−1 ,

qi,i−1 = q τi−1 .

This is called the (symmetric) trap model with drift. We are mostly concerned with limit theorems for these processes in the case of vanishing drift. We therefore suppose that µ ≥ 0, β ≥ 0 and X (N) = (Xt(N) : t ≥ 0) is defined by Xt(N ) = XN t , where X = (Xt : t ≥ 0) is a trap model with jump probabilities µ µ 1 1 (N) (N ) (N) 1+ β , q = 1−p = 1− β . p = 2 N 2 N

(We take µ ≤ 1 if β = 0.). We define the following limiting processes:

• Inverse stable subordinator. For 0 < α < 1 the stable subordinator is the increasing Lévy process (Subt : t ≥ 0) with E e−λSubt = exp − t Γ(1 − α) λα . Its right-continuous inverse (Sub−1 s : s ≥ 0) defined by

Sub−1 s = inf{t > 0 : Subt > s} is the inverse stable subordinator with index α. • Fontes Isopi Newman diffusion with drift µ. Suppose (B(t) : t ≥ 0) is a Brownian motion with drift µ and (ℓ(t, x) : t ≥ 0, x ∈ R) its local times. Let ρ be an independent stable measure with index 0 < α < 1, defined as the random measure whose cumulative distribution function is a twosided stable subordinator with the same index. Define an increasing function Z φ(t) = ℓ(t, x) ρ(dx), 3

and its inverse ψ(s) = inf{t > 0 : φ(t) > s}. Then (Finµs : s ≥ 0) given by

Finµs = B(ψ(s))

is a Fontes Isopi Newman diffusion with drift µ. We always denote by “=⇒” convergence in distribution, averaging over the trap environment. Theorem 2.1 (Scaling limits). We have the following limit laws, where “=⇒” denotes convergence in distribution on the Skorokhod space D[0, 1] of right-continuous functions with left-hand limits. (a) If 0 ≤ β
0 then X (N ) µα =⇒ Sub−1 . N α(1−β) Γ(1 + α)

(b) If β =

(c) If β >

α α+1

and µ > 0 then

α α+1

or µ = 0 then

X (N) =⇒ Finµ . β N X (N ) =⇒ Fin0 . α α+1 N

The scaling limit in regime (c) has been identified by Fontes et al. (2002) in the case of the trap model without drift (µ = 0); the inverse stable subordinator has been observed, using methods different from ours, as a scaling limit in trap models with constant drift (β = 0) by Zindy (2009). Monthus (2004) has some interesting results for the asymptotics α ↓ 0. The diffusion with drift, which we observe in the critical regime, represents a previously unobserved scaling behaviour. α Theorem 2.2 (Ageing in the presence of slowly vanishing drift). If 0 ≤ β < α+1 and µ > 0, then there exist nonnegative nondegenerate random variables ξt such that

τX (N) t

N 1−β

=⇒ ξt .

Define a function c(t) ∈ (0, 1) by c(t) = E[exp{− ξ1t }]. Then we have (N) lim P Xt(N ) = Xt+s for all 0 ≤ s ≤ N −β = c(t) . N ↑∞

Remark 2.3. Note that the ageing exponent defined in the introduction equals γ = 1−β in this case. The limit variable ξt describes the traps from the point of view of the particle. We define two independent series of nonnegative i.i.d. random variables U1 , U2 , . . . and S1 , S2 , . . . such that 4

• Si is the product of two independent random variables, a Pareto variable with index α and an exponential variable with mean 1/µ; • Ui is a random variable with (for some constant c depending only on α) P {Ui > x} ∼ c

µα xα

as x ↑ ∞,

and the law of ξt can be described as j−1 j ∞ nX o X X t P P ξt > v = (Ui + Si ) + Uj < ≤ (Ui + Si ) , v j=1 i=1 i=1

for v > 0.

Here, loosely speaking, the variables Si represent periods in which the walker is in deep traps, while the Ui represent the travel times between these traps. We also have results in the rapidly vanishing and critical drift regimes. Theorem 2.4 (Ageing in the presence of rapidly vanishing drift). If β ≥ exist nonnegative nondegenerate random variables ζt such that τX (N) t

1

N α+1

α , α+1

then there

=⇒ ζt .

Define a function k(t) ∈ (0, 1) by k(t) = E[exp{− ζ1t }]. Then we have (N ) lim P Xt(N) = Xt+s

N ↑∞

α for all 0 ≤ s ≤ N − α+1 = k(t) .

1 Remark 2.5. In this regime the ageing exponents equals γ = α+1 . We observe a joint convergence of the rescaled process and the rescaled trap environment interpreted as a random measure, X α 1 − α+1 − α+1 (N ) α N τz δN − α+1 z =⇒ (Finθ , ρ) , X ,N z∈Z

where θ = µ if β = given as

α , α+1

and θ = 0 otherwise. The limiting random variable ζt is then ζt = ρ(Finθt ).

Remark 2.6. In both of our ageing results, unless β = 0, the time typically spent by the process in the current state is a sublinear function of time. This kind of phenomenon is sometimes called sub-ageing and is exhibited by an ageing exponent γ < 1.

5

3

Proof of Theorem 2.1 (a)

The basic idea of the proof to show that the process X (N) is mostly increasing and therefore essentially invertible. The convergence of one-dimensional marginals will then be proved for the inverse process using Laplace transforms. This will be extended to finite-dimensional marginals using asymptotic independence properties, and finally to Skorokhod space by verification of a continuity criterion. The first lemma ensures that X (N ) is mostly increasing. We denote by Tx the first time where the process X hits level x > 0. We can write Tx =

∞ X n=0

ηn τSn 1 max Sj < x , j≤n

where (Sn : n ≥ 0) is the random walk embedded in the process X and (ηn : n ≥ 0) an independent family of independent standard exponential random variables. Lemma 3.1. There exists a constant C such that, for x large enough, n o x P sup sup Xu(N) − Xv(N) ≥ x ≤ C tα N α exp − µ β , N v≤t u≤v

for all t > 0.

Proof. It is clear that n o sup sup Xu(N) − Xv(N ) ≥ x v≤t

u≤v

∞ n o [ ⊆ Tj < Nt; Xv ≤ j − x for some v > Tj . j=0

Furthermore, n o n o Tj < Nt; Xv ≤ j − x for some v > Tj ⊆ Tj < Nt; min Sk(j) ≤ −x , k≥1

where (Sk(j) : k = 1, 2, . . .) is the random walk embedded in (Xv − j : v ≥ Tj ), which is independent of (Xv : v ≤ Tj ). Consequently, n

(N)

(N )

P sup sup Xu − Xv v≤t

u≤v

∞ o n oX n o ≥ x ≤ P min Sk ≤ −x P Tj < Nt . k≥1

j=0

For the first term, n o q (N ) x 1 − µN −β x x . ≤ exp − µ P min Sk ≤ −x = (N ) = k≥1 p 1 + µN −β Nβ 6

(2)

We next note that

j−1 ∞ ∞ o nX X X τk ηk < Nt . P Tj < Nt ≤ P j=0

j=0

k=0

Noting that the tail of τ0 η0 is regularly varying with index α and using the renewal theorem for this class of random variables, see e.g. Erickson (1970), we see that the sum on the right is bounded by C(Nt)α . This completes the proof of the lemma. A direct consequence of Lemma 3.1 is the following limit in probability. Lemma 3.2. If 0 ≤ β
0 we define σz := min{k ≥ 1 : Sk = z} and Az := {Sk > 0 for all k > σz }.

(11)

Then E ψ

ℓ(x) (0) Nβ

ψ

ℓ(x) (z) Nβ

(x) = E ψ ℓ N β(0) ψ =: E1 + E2 .

ℓ(x) (z) ; A β z N

+E ψ

ℓ(x) (z) ℓ(x) (0) c ψ ; A β β z N N

Using the Cauchy-Schwarz inequality, we obtain E2 ≤ P1/2 (Acz )E1/2 ψ 2 ≤ P1/2 (Acz )E1/4 ψ 4 ≤ P1/2 (Acz )E1/2 ψ 4

ℓ(x) (0) 2 ℓ(x) (z) ψ Nβ Nβ ℓ(x) (0) 1/4 4 ℓ(x) (z) E ψ Nβ Nβ (x) ℓ (0) . Nβ

Noting that P(Acz ) = P{mink≥1 Sk ≤ −z} and applying (2), we get E2 ≤ exp − µ 2Nz β E1/2 [ψ 4

ℓ(x) (0) ]. Nβ

(12)

Since ℓ(x) (0) = ℓ(z) (0) on the event Az , using the Markov property, E1 = E ψ =E ψ ≤E ψ ≤E ψ

Combining (12) and (13) gives Cov ψ

ℓ(x) (0) Nβ

,ψ

ℓ(z) (0) ℓ(x) (z) ψ ; A z β β N N ℓ(x−z) (0) ℓ(z) (0) E ψ ; min Sk Nβ Nβ k≥1

> −z

(x−z) ℓ(z) (0) E ψ ℓ N β (0) Nβ ℓ(x) (0) ℓ(z) (0) E ψ Nβ . Nβ ℓ(x) (z) Nβ

≤ exp − µ 2Nz β E1/2 ψ 4

10

(13)

ℓ(x) (0) Nβ

.

Therefore, Var

x−1 hX

ψ

i

ℓ(x) (z) Nβ

z=0

=

x−1 X

ℓ(x) (z) Nβ

Var ψ

z=0

≤ xE ψ 2 1/2

≤ xE

ℓ(∞) (0) Nβ

β

≤ CxN E

4 ψ

Cov ψ

ℓ(x) (y) Nβ

z=0 y=z+1

+2

x−1 x−1 X X

z=0 y=z+1

(∞) ψ 4 ℓ N β(0) 1/2

+2

x−1 X x−1 X

1/2

+ xE

ℓ(∞) (0) . Nβ

,ψ

ℓ(x) (z) Nβ

1/2 4 y−z exp − µ 2N E ψ β

(x) ψ 4 ℓ N β(0)

∞ X z=0

ℓ(x) (0) Nβ

exp − µ 2Nz β

¿From this bound and Chebyshev’s inequality we get x−1 n X P ψ

ℓ(x) (z) Nβ

z=0

−E

x−1 X

ψ

z=0

Applying (7) and (8), we have x−1 n X P ψ z=0

ℓ(x) (z) Nβ

−E

On the other hand, Eψ

ℓ(x) (z) Nβ

= Eψ

ℓ(x) (z) Nβ

ℓ(x−z) (0) Nβ

x−1 X

Px−1 z=0

≥E ψ

= Eψ

Consequently, ℓ(x) (z) Nβ

z=0

ℓ(x) (z) Nβ

ℓ(∞) (z) Nβ

o CN β > εx ≤ 2 . εx

= Eψ

ℓ(∞) (0) . Nβ

ℓ(∞) (0) ; Ax−z Nβ ℓ(∞) (0) c ℓ(∞) (0) − E ψ N β ; Ax−z β N (∞) (∞) ℓ (0) − P1/2 (Acx−z )E1/2 ψ 2 ℓ N β(0) . Nβ

− xE ψ

o CxN β E1/2 [ψ 4 ( ℓ(∞) (0) )] Nβ . > εx ≤ 2 2 εx

ψ(ℓ(x) (z)/N β ). On the one hand,

≤ Eψ

≥ Eψ

x−1 X ψ E

ψ

z=0

We now estimate the expectation E Eψ

ℓ(x) (z) Nβ

ℓ(∞) (0) Nβ

1/2

≤E

(∞) ψ 2 ℓ N β(0)

∞ X z=0

P1/2 Acz .

Applying (2) and (8), we conclude that 1/2

lim supE N→∞ x/N β →∞

(∞) ψ 2 ℓ N β(0)

≤ sup E1/2 ψ 2 N ≥1

∞ 1 X 1/2 c P (Az ) × x z=0

ℓ(∞) (0) Nβ

∞ 1 X exp{−µ 2Nz β } = 0. x β x/N →∞ z=0

× lim sup 11

(14)

Using also (7), we infer that lim

N→∞ x/N β →∞

x−1 1 X E ψ x z=0

ℓ(x) (z) Nβ

= lim E ψ

ℓ(∞) (0) Nβ

N →∞

=

Z

∞

ψ(y)µe−µy dy.

(15)

0

Combining (10), (14) and (15) finishes the proof of the lemma. We now start with the proof of Theorem 2.1 (a), first considering one-dimensional marginals. Using Lemma 3.3 for x = y = aN α(1−β) we have n X ℓ(x) (z) α o λα E exp − Nλ TaN α(1−β) = E exp − Γ(1 − α) N α(1−β) (1 + o(1)) . Nβ z vN 1−β t

for v > 0.

Our strategy is to look at the deep traps z ∈ Z defined by τz > vN 1−β and at the ‘travelling intervals’ defined as the times which the process X spends travelling from one deep trap to the next one to its right. We show that during the intermediate intervals, which seperate the travelling intervals, the process spends most of its time in a deep trap. The length of travelling intervals and intermediate intervals are both of order N and we determine the asymptotic distribution of their lengths, which enables us to find the limit above. The further statements in Theorem 2.2 follow easily from this. Define the sequence of deep traps (with x0 = 0), as xj = min{z > xj−1 : τz > vN 1−β } for j ≥ 1. The following lemma reveals the typical distance of two successive traps. Lemma 6.1. For any u > 0 and j ≥ 0, we have α lim P xj+1 − xj ≥ uN α(1−β) = e−u/v . N →∞

16

Proof. Using the tail behaviour of the random variables τz given by (1), we have for any r > 0, 1 + o(1) α r , P {xj+1 − xj ≥ r} = 1 − vN 1−β from which the result follows by Euler’s formula. Next, we investigate the time spent in a deep trap before the next deep trap is hit for the first time. Lemma 6.2. For all j ≥ 0, ℓ(xj+1 ) (xj ) =⇒ ξ, Nβ where ξ is exponentially distributed with mean 1/µ if β > 0, and geometrically distributed with mean 1/µ if β = 0. Proof. Recall (11) and let Ax,y = Sk > x for all k ≥ σy for x < y.

Keeping x1 , x2 , . . . fixed by conditioning on the trap environment, we have ℓ(xj+1 ) (xj ) ℓ(∞) (xj ) ℓ(∞) (xj ) ≥ ≥ 1Axj ,xj+1 . Nβ Nβ Nβ Observe that ℓ(∞) (xj ) is geometrically distributed with success parameter µ/N β . Therefore ℓ(∞) (xj ) =⇒ ξ, Nβ where ξ is exponentially distributed with parameter µ. It therefore suffices to show that ℓ(∞) (xj ) 1Acxj ,xj+1 Nβ converges weakly to zero. As the first factor converges, it further suffices to show that P(Acxj ,xj+1 ) converges to zero. By (2) we have P

Acxj ,xj+1

≤ E exp

n

−

µ(xj+1 −xj ) Nβ

o .

Averaging over the trap environment and using Lemma 6.1 together with the fact that α(1 − β) > β, we see that the right hand side converges to zero. Lemma 6.3. For τ as in (1), we have as y ↑ ∞ and λ/y ↓ 0, Z ∞ −λτ −α α −λz α y E e |τ ≤y =1+y 1 − Γ(1 − α)λ − e z α+1 dz + o (λ + 1)α y −α . 1

17

Proof. We have P {τ ≤ y} = 1 − y −α + o(y −α ), and hence 1 = 1 + y −α + o(y −α). P {τ ≤ y} Moreover, using integration by parts, Z ∞ −λτ λ y E e ;τ > y = e− y x P {τ ∈ dx} y Z ∞ λ −λ λ P {τ > x}e− y x dx = e P {τ > y} − y Zy ∞ = e−λ P {τ > y} − λ P {τ > yz}e−λz dz. 1

As, for all z > 1, P {τ > yz} −→ z −α , P {τ > y} we obtain, by dominated convergence, R∞ Z ∞ P {τ > yz}e−λz dz 1 −→ e−λz z −α dz. P {τ > y} 1 Therefore

τ −λ y

E e

Z ∞ h i −λ ; τ > y = P {τ > y} e − λ e−λz z −α dz (1 + o(1)) Z ∞ 1 −α α = y (1 + o(1)) e−λz z α+1 dz.

1

λ

Recalling that Ee− y τ = 1 − Γ(1 − α)( λy )α + o(λα y −α) and summarising, λ E e− y τ | τ ≤ y λ λ 1 Ee− y τ − E e− y τ ; τ > y = P {τ ≤ y} Z ∞ −α α −λz α =1+y 1 − Γ(1 − α)λ − e z α+1 dz + o (λ + 1)α y −α . 1

This completes the proof.

We now define quantities, which will be shown to converge in distribution to the families U1 , U2 , . . . and S1 , S2 , . . . described in Remark 2.3. Fix j ≥ 1 and let S (j) = (Sn(j) : n = 0, . . . , ζ (j) ) be the embedded random walk started from the first hitting of xj−1 and stopped upon hitting xj , such that S0(j) = xj−1 and Sζ(j)(j) = xj . Let ℓj (x) be the local time in x of the embedded random walk and let 0 = n1 < n2 < · · · < nm < ζ (j) 18

with m = ℓj (xj−1 ) be the complete list of visits to xj−1 by S (j) . Define ζ (j) −1 (N )

Uj

X

=

i

i=nm +1

and

(j)

τS (j) ηi Si

ℓj (xj−1 )

X

(N)

Sj−1 = τxj−1

ηni (xj−1 ),

i=1

where (ηi (x) : i ∈ N, x ∈ Z) is a family of independent standard exponential variables, independent of everything else. Observe that, roughly speaking, Uj(N ) /N is the time the (N ) /N the time spent in the trap xj−1 process X (N) requires to travel from xj−1 to xj and Sj−1 before the first hit of xj . (N ) It is important to note that Uj(N ) is independent of U1(N ) , . . . , Uj−1 and of S1(N) , . . . , Sj(N) , (N) and also Sj(N ) is independent of S1(N ) , . . . , Sj−1 .

For 1 ≤ l ≤ ℓj (xj−1 ) − 1 we define nl+1 −1 (N )

Qj,l =

X

n=nl +1

τSn(j) ηn Sn(j) ,

(N ) such that Qj,l /N is the time spent by X (N) in the l-th excursion from xj−1 to xj−1 before reaching xj . Further define the set

(N )

Rj

ℓj (xj−1 )−1

=

[

τxj−1

i=1

i X

ηnl (xj−1 ) +

i−1 X

(N)

Qj,l , τxj−1

l=1

l=1

i X l=1

ηnl (xj−1 ) +

i X l=1

(N )

Qj,l .

) The set N1 R(N is the (random) set of times which X (N) spends in excursions from xj−1 j (either to the left or to the right) which return to xj−1 . We first show that the time spent in these excursions is negligible. ) Lemma 6.4. Let Rj(N ) = |R(N j |. Then

(a)

Rj(N) ⇒ 0 as N ↑ ∞; N

(b) for every t > 0, we have lim P Nt ∈ Rj(N ) = 0. N ↑∞

Proof. If (b) holds, then ERj(N ) = consequence of (b).

RN (N ) P t ∈ R dt = o(N), hence (a) is an immediate j 0

It remains to show (b). By Lemma 6.1 the distance between xj−2 and xj−1 is of order N α(1−β) . Then, using (2), we conclude that the probability that X hits xj−2 after hitting xj−1 converges to zero as N → ∞. Consequently, ) P t ∈ Rj(N ) = P {t ∈ R(N j } ∩ Aj + o(1), 19

) where Aj = {Xt(N ) > xj−2 for all t > Uxj−1 }. It follows from the definition of Q(N j,l , that, conditioned on (τz : z ∈ Zd ),

ℓj (xj−1 )−1

(N )

P {t ∈ Rj } ∩ Aj = E

X

P

k n X

Ui + τxj−1 ηk+1 (xj−1 ) < Nt
0. (17) H(x + y) − H(x) ≤ min 1, τxj−1 τxj−1 τxj−1 (We postpone the derivation of this inequality to the end of the proof.) Using this bound we get n Qe(N) o e (N) z−Q P {t ∈ Rj(N ) } ∩ Aj ≤ 2E τxj . + P η(xj−1 ) ∈ τx j , τx z j−1

Noting that

we have

n P η(xj−1 ) ∈

j−1

(N)

e z−Q j τxj−1

, τx

z j−1

o

Qe(N) ≤ E τxj , j−1

) Qe(N ) j P {t ∈ R(N } ∩ A ≤ 3E . j j τx j−1

Recalling also that τxj−1 ≥ vN

1−β

, we arrive finally at the bound

) P {t ∈ R(N j } ∩ Aj ≤ 20

3 e(N ) . EQ j 1−β vN

j−1

Going back to the unconditioned probability, we have, uniformly in t, P t ∈ Rj(N ) ≤

3 e(N ) + o(1). EQ j vN 1−β

It follows from (1) that E[τz | τz ≤ vN 1−β ] ≤ CN (1−α)(1−β) . Then, e(N ) ≤ CN (1−α)(1−β) E[L − 1; L < ∞], EQ j

where L is the length of an excursion of the embedded random walk. As P{L = k} = 0, if k is odd, and, with p = p(N) , (see e.g. III.9 in Feller (1968)) 1 2j − 2 j P{L = k} = p (1 − p)j , 8j j − 1 if k = 2j is even, we obtain ∞ X

(k − 1) P{L = k} =

∞ X 2j − 1 2j − 2 j=1

k=1

≤

∞ X 2j j=0

Note that 2p − 1 =

µ . Nβ

j−1

8j j

p(1 − p)

j

pj (1 − p)j = E[ℓ(∞) (0)] =

1 . 2p − 1

Thus, E[L − 1; L < ∞] ≤ CN β . Hence, we have e(N ) ≤ CN β+(1−α)(1−β) = o(N 1−β ). EQ j

Therefore, it remains to prove (17). P Let θx denote the first time when the random walk n1 Vi leaves the interval (−∞, x), that is, n X θx := min{n ≥ 1 : Vi ≥ x}. 1

Then, for every y > 0, P

θx nX i=1

o

Vi ≤ x + y =

Z

x

P 0

x −1 n θX

i=1

o Vi ∈ du P V1 ∈ (x − u, x + y − u) ≤ y sup f (z), z>0

where f is the density of V1 . This function is the convolution of densities of τxj−1 η and ) Q(N j . Then supz>0 f (z) does not exceed the maximal value of the density of τxj−1 η, which is equal to 1/τxj−1 . Consequently, P

θx nX i=1

o

Vi ≤ x + y ≤ min 1,

21

y τxj−1

.

(18)

Using the Markov property, we obtain ∞ θx θx x +k h nX o n θX oi X X H(x + y) − H(x) = E 1 Vi ≤ x + y 1 + 1 Vi Vi ≤ x + y − i=1

k=1

i=1

i=θx +1

θx θx h nX i o X =E 1 Vi Vi ≤ x + y H x + y − i=1

i=1

≤ H(y)P

θx nX i=1

o Vi ≤ x + y .

We now note that from the definition of V1 follows the inequality ∞ k nX o X H(y) ≤ 1 + P ηi ≤ yτxj−1 ≤ 1 + k=1

i=1

y τxj−1

.

Combining this with (18), we arrive at (17). This completes the proof of the lemma. Lemma 6.5. For N ↑ ∞, Sj(N) =⇒ Sj , vN where Sj is the product of two independent random variables, a Pareto variable with index α, and an exponential variable with mean 1/µ. Proof. First note that, for y ≥ v and x ∈ Z, v α 1−β 1−β = lim P τx > yN τx > vN . N ↑∞ y

Therefore we have, for all j ∈ N,

We write

v α 1−β 1{y ≥ v}. lim P τxj > yN = N ↑∞ y ℓj+1 (xj ) X Sj(N) τxj ℓj+1 (xj ) 1 ηi (xj ) = vN vN 1−β Nβ ℓj+1 (xj ) i=1

(19)

and observe convergence of all three factors on the right hand side. Indeed, the first factor converges in distribution to a Pareto law, by (19), and the second factor to an exponential law with mean 1/µ, by Lemma 6.2. Moreover, the second factor is independent of the first. To understand the third factor, recall from the discussion of the second factor that ℓj+1(xj ) converges to infinity in probability. Thus, by the weak law of large numbers, the third factor converges to one in probability. Hence, the product Sj(N ) /vN converges to the product of an independent Pareto and exponential law.

22

Lemma 6.6. For N ↑ ∞, Uj(N ) =⇒ Uj , vN where Uj is a random variable with P{Uj > x} ∼ c

µα xα

as x ↑ ∞,

for some c > 0 depending only on α. Proof. Recall that ζ (j) −1

Uj(N ) =

X

(j)

τS (j) ηi Si i

i=nk +1

.

Conditional on xj−1 , xj the random variables τx , x = xj−1 + 1, . . . , xj − 1, are still independent with τx conditioned to satisfy τx ≤ vN 1−β . We first consider the case β > 0. Writing Laplace transforms (N) Uj E exp − λ vN

∼ EE h =E =E

h

xj −1

Y

E exp −

x=xj−1 +1 xj −1

Y

x=xj−1 +1 xj −1

Y

x=xj−1 +1

ℓj (x)

E 1+

λ vN

X

τx

λτx −ℓj (x) vN

h E exp −

ηi (x)

i=1

i

| τx ≤ vN 1−β

λ τ ℓ (x) vN x j

i

1 + O(N −β )

i 1−β τ ≤ vN , x

where in the last step we have used that, on the event {τx ≤ vN 1−β }, −ℓj (x) λτx x 1 + λτ = exp − ℓ (x) log 1 + j vN vN λ τx ℓj (x) 1 + O(N −β ) . = exp − vN By Lemma 6.3, we have E exp −

for

1 + O(N −β ) τx ≤ vN 1−β 1 + o(1) ℓj (x) = 1 + α(1−β) α ψλ N . β N v λ τ ℓ (x) vN x j

α α

ψλ (y) = (1 − Γ(1 − α))λ y −

Z

1

By Lemma 3.4 and (10) we have xj −1 1 xj −xj−1

X

x=xj−1

ℓj (x) ψλ N β

23

⇒

Z

∞ α e−λyz z α+1 dz.

ψλ (y) µe−µy dy.

Altogether, (N) Uj lim E exp − λ vN N ↑∞ Z ∞ n o xj −xj−1 = lim E exp vα N α(1−β) ψλ (y) µe−µy dy (1 + o(1)) N ↑∞ 0 Z ∞ n Z ∞ o = e−u exp u ψλ (y) µe−µy dy du 0 0 Z ∞Z ∞ −1 Z ∞ α α −µy α dz µe−µy dy = Γ(1 − α)λ y µe dy + e−λyz z α+1 0 0 1 Z ∞ −1 α dz . = sinαπαπ µλ + α z α+1 (1+z λ ) µ

1

The limit is continuous at λ = 0 and hence, by Bochner’s theorem, see e.g. Theorem 5.22 in Kallenberg (2002), it is the Laplace transform of some random variable Uj . Theorem 4 in XIII.5 of Feller (1971) implies the statement about the tail behaviour. Pℓj (x) Assume now that β = 0. Set θj (x) = i=1 ηi (x). Then (N) Uj E exp − λ vN j −1 h xY ∼ EEE

x=xj−1 +1

h = EE

xj −1

Y

x=xj−1 +1

exp −

λ vN

E exp −

τx θj (x)

λ vN

i

i τx θj (x) | τx ≤ vN ,

Using Lemma 6.3 once again, we get i 1 + o(1) ψ θ (x) . τx θj (x) | τx ≤ vN = 1 + λ j N αvα Repeating the arguments of Lemma 3.4, one can easily see that E exp −

λ vN

xj −1 1 xj −xj−1

X

x=xj−1

ψλ θj (x) ⇒ Eψλ (θ),

Pℓ(∞) (0) ηi (0). Since ℓ(∞) (0) is geometrically distributed, θ is exponentially where θ = i=1 distributed with mean 1/µ. Thus, Z Eψλ (θ) = ψλ (y) µe−µy dy. This means that the remaining part of the proof coincides with that for the case β > 0. ) Recall that Rj(N ) = |R(N j |. Then

1 X (N ) (Ui + Si(N) + Ri(N) ) + Uj(N ) N i=1 j−1

24

is the total time X (N) takes to hit xj . For the lower bound in Theorem 2.2 we use Lemmas 6.4, 6.5, and 6.6, and get, for any M > 0, n o P τX (N) /N 1−β > v t

≥

M X

P

j=1

j−1 nX

(Ui(N ) + Si(N) + Ri(N) ) + Uj(N )

i=1

< Nt ≤

j X i=1

o ) (Ui(N ) + Si(N ) + Ri(N ) ) − P Nt ∈ R(N j

j M o nX X X t (Ui + Si ) , P −→ (Ui + Si ) + Uj < ≤ v i=1 j=1 i=1 j−1

and we get the required lower bound by letting M ↑ ∞. For the upper bound, we have, for any M > 0, n o P τX (N) /N 1−β > v

as N ↑ ∞,

t

≤

M X

P

j=1

j−1 nX

(N )

(Ui

(N )

+ Si

+ Ri ) + Uj

j=1

< Nt ≤

P

j−1 nX

j X

(N )

(Ui

(N )

+ Si

(N )

+ Ri )

i=1

M nX i=1

−→

(N )

i=1

+P M X

(N )

o (N ) (Ui(N ) + Si(N) + Ri(N ) ) + UM < Nt +1

+P

M nX

o X t (Ui + Si ) ≤ v i=1 j

(Ui + Si ) + Uj
0 there exists an open ball B ⊂ R with sup P Y [ν (N ) , S (N ) ](t) 6∈ B < ε . N ≥1

Now given 0 < u < v we let G = B × (u, v) and assume that u, v are not weights of atoms of ν. With the notation from above we have lim P ν (N ) ({Y [ν (N ) , S (N) ](t)}) ∈ (u, v), Y [ν (N ) , S (N) ](t) ∈ B N →∞

= lim

N →∞

l X

P{Y [ν

(N )

,S

(N)

(N )

](t) = xi } =

i=1

l X i=1

P{Y [ν, id](t) = xi }

= P ν({Y [ν, id](t)}) ∈ (u, v), Y [ν, id](t) ∈ B ,

which completes the proof as ε > 0 was arbitrary. 26

By a classical stable limit theorem, see Proposition 3.1 in Fontes et al. (2002), there exists a coupling of the measures ν (N ) in the proof of Theorem 2.1 (b) and (c) such that, almost surely, ν (N ) converges to ρ in the point process sense. Obviously, S (N) converges to the identity uniformly on compact sets, and hence Lemma 7.1 shows that lim

N →∞

τX (N) t

N

1 α+1

= lim ν (N ) ({Y [ν (N ) , S (N) ](t)}) = ρ(Y [ρ, id])(t) = ρ(Finθt ) N →∞

in law.

The ageing result follows by the same argument as in Theorem 2.2. Acknowledgements: The second author was supported by an Advanced Research Fellowship of the Engineering and Physical Sciences Research Council (EPSRC). The work was started when the first and the third author visited the Department of Mathematical Sciences, University of Bath, continued when the second author visited the Technische Universität M¨ unchen, and it was completed when the second and third author visited the Isaac Newton Institute at Cambridge. All this support is gratefully acknowledged.

References ˇ y (2009). Convergence to fractional kinetics for random walks M. Barlow, and J. Cern´ associated with unbounded conductances. Probab. Theory and Relat. Fields (to appear). G. Ben Arous, A. Bovier, and V. Gayrard (2003). Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys. 236, 1–54. ˇ y (2006). Dynamics of trap models. In Ecole d’Eté de Physique G. Ben Arous, and J. Cern´ des Houches LXXXIII, “Mathematical Statistical Physics”, pp. 331–394. Amsterdam: North-Holland. ˇ y (2007). Scaling limits for trap models on Zd . Ann. Probab. 35, G. Ben Arous, and J. Cern´ 2356–2384. ˇ y and T. Mountford (2006). Aging in two-dimensional Bouchaud’s G. Ben Arous, J. Cern´ model. Probab. Theory and Relat. Fields 134, 1–43. G. Ben Arous, A. Dembo and A. Guionnet (2001). Aging of spherical spin glasses. Probab. Theory and Relat. Fields 120, 1–67. P. Billingsley (1999). Convergence of probability measures (Second ed.). Wiley Series in Probability and Statistics: Probability and Statistics. New York: John Wiley & Sons Inc. A Wiley-Interscience Publication. A. N. Borodin, and P. Salminen Handbook of Brownian Motion – Facts and Formulae, Birkhäuser Verlag, Basel-Boston-Berlin, 1996. J.-P. Bouchaud (1992). Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713.

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K. B. Erickson (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263-291. W. Feller (1968). An introduction to probability theory and its applications I. New York: John Wiley & Sons Inc. W. Feller (1971). An introduction to probability theory and its applications II. New York: John Wiley & Sons Inc. L. R. G. Fontes, M. Isopi, and C. M. Newman (2002). Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2), 579–604. O. Kallenberg (2002). Foundations of modern probability. New York: Springer Verlag. C. Monthus (2004). Nonlinear response of the trap model in the aging regime: exact results in the strong disorder limit. Phys. Rev. E 69, 026103. P. Mörters, M. Ortgiese, and N.A. Sidorova (2009). Ageing in the parabolic Anderson model. Preprint arXiv:0910.5613. C. Stone (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7, 638–660. O. Zindy (2009). Scaling limit and aging for directed trap models. Markov Processes and Related Fields 15, 31–50. Nina Gantert Institut f¨ ur Mathematische Statistik Universität M¨ unster Einsteinstr. 62 D-48149 M¨ unster, Germany [email protected] Peter Mörters Department of Mathematical Sciences University of Bath Claverton Down Bath BA2 7AY, United Kingdom [email protected] Vitali Wachtel Mathematisches Institut der Universität M¨ unchen Theresienstr. 39 D-80333 M¨ unchen, Germany [email protected]

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