Asymptotically one-dimensional diffusions on scale-irregular gaskets.

Asymptotically one-dimensional diusions on scale-irregular gaskets. Tetsuya Hattori

Faculty of Engineering, Utsunomiya University, Ishii-cho, Utsunomiya 321, Japan e-mail: [email protected]

April 25, 1997 Abstract

A simple class of fractals which lack exact self-similarity is introduced, and the asymptotically one-dimensional diusion process is constructed. The process moves mostly horizontally for very small scales, while for large scales it diuses almost isotropically, in the sense of the o-horizontal relative jump rate for the decimated random walks of the process. An essential step in the construction of diusion is to prove the existence of appropriate time-scaling factors. For this purpose, a limit theorem for a discrete-time multi-type supercritical branching processes with singular and irregular (varying) environment, is developed. Mathematics Subject Classi cation: 60J60, 60J25, 60J85, 60J15

Running title: Asymptotically one-dimensional diusion on scaleirregular gasket

1 Introduction. In this paper we construct asymptotically one-dimensional diusion processes on the scale-irregular abb-gaskets. We generalize the Sierpinski gasket, a triangle based fractal, to de ne the scale-irregular abb-gaskets, which we introduce as simple examples of nitely rami ed fractals that are scale-irregular, i.e. do not have exact self-similarity. Let f(aN ; bN ); N 2 Zg be a sequence of pair of positive integers. Let H00 be the set of vertices and edges of a triangle, considered as a graph. Join a0 + 2b0 H00 s' as in Figure 1 to form a triangle H?0 1 , so that one side of H?0 1 is composed of a0 + 1 H00 s' and the other two sides are composed of b0 + 1 H00 s'. Repeat this procedure recursively to obtain H?0 n?1 1

from a?n + 2b?n H?0 n s', for n 2 Z+ . We regard one of the three vertices of H?0 n to be the origin O and regard H?0 n  H?0 n?1 . Let H0 be the two copies of the in nite graph (extending in nitely outward)

1 [

n=0

H?0 n joined at the origin

O. Next, de ne H1  H0 by adding a structure (bonds and vertices) similar to the structure of H?0 1 in Figure 1, but with (a0 ; b0) replaced by (a1 ; b1 ), to

each of the unit triangle. Repeat this procedure recursively inwards and add substructure of aN +1 +2bN +1 triangles to each of the smallest triangle in HN to obtain HN +1  HN . We call the graph HN a pre-gasket (of scale N ). Let GN denote the set of vertices in HN . Each element of GN (including O) has four N -neighbor points; by an N -neighbor point we mean anvertex in HN joined by Y an edge of HN . If N > N 0 then GN  GN 0 . Put n = minfak +1; bk +1g?1, k=1

n 2 Z+ . We introduce metric d such[that d(x;[ y) = n if x and y are n-neighbor points. See Appendix A. G = GN = HN is the scale-irregular abbN

N

gasket. The de nitions are independent of embedding into Euclidean spaces, but for simplicity, the terminology for subsequent explanations will be as if a scale-irregular abb-gasket G is embedded in R2 such that x-axis is contained in G and G is contained in upper half plane (e.g. horizontal means parallel to x-axis). If aN and bN are independent of N , the corresponding fractal has exact self-similarity. The Sierpinski gasket is the scale-irregular abb-gasket with aN = bN = 1; N 2 Z. If bN = aN , N 2 Z, which we call the scale-irregular aaa-gaskets, there is an obvious local spatial symmetry (isotropy) shared with Sierpinski gasket. For a process X taking values in G we de ne Tn;i (X ) ; n 2 Z , (the time that X hits Gn for the i +1-th time, counting only once if it hits the same point more than once in a row) by Tn;0(X ) = inf ft  0 j X (t) 2 Gn g , and

Tn;i+1 (X ) = inf ft > Tn;i (X ) j X (t) 2 Gn n fX (Tn;i(X ))gg ; i = 0; 1; 2;


: For an integer n and a Markov process X on G or on GN for some N  n, we call a simple random walk X (n) on Gn de ned by X (n)(i) = X (Tn;i (X )) the n-decimated walk of X . By de nition, Proposition 1.1. If n < N and X (n) and X (N ) are the n and N -decimated walks of X respectively, then X (n) is the n-decimated walk of X (N ). For N 2 Z and w > 0, we de ne a simple random walk XN;w on GN as follows. At each integer time, the random walker jumps to one of the four N -neighbor points, and the relative rates of the jumps are 1 (resp. w) for a jump in horizontal (resp. 60 or 120) direction. We prove in this paper the following. 2

Theorem 1.2. Assume that f(aN ; bN ); N 2 Zg is a bounded sequence of pairs of integers satisfying aN  2; bN  2; bN < 2aN ; N 2 Z; and let G be the scale-irregular abb-gasket de ned by this sequence. Then there exist a sequence of positive numbers wN , N 2 Z satisfying (1.2) lim w = 0; N !1 N

(1.1)

and a symmetric Feller process X (a continuous G-valued strong Markov process whose transition operators map the space of bounded continuous functions R into itself, respect to a measure  on G de ned by f d = QN symmetric ?with  P lim (a + 2bk ) 1 x2GN f (x)), such that for N 2 Z, the N -decimated N !1 k=0 k walk of X is equal in law to the random walk XN;wN . The assumptions (1.1) are to avoid complications. We will prove Theorem 1.2 for any w0 satisfying

(1.3)

w0 2 I def = (0; inf f2aN =bN g ? 1): N

From Proposition 1.1 and Theorem 1.2, the (N ? 1)-decimated walk of XN;wN is equal in law to XN ?1;wN ?1 , from which it follows that the sequence fwN g satis es a recursion relation (1.4)

wN ?1 = f(aN ;bN ) (wN ) ; N 2 Z;

where (1.5)

+ a)b + (a b + a + b)wg f(a;b) (w) = b(2bw+f(1 2) + 2(b2 + a + b)w + b2w2 :

Proof of (1.4) is elementary (but lengthy), and is similar to that of [11, Proposition 1.1]. It is elementary to see that Proposition 1.3. If (1.1) and (1.3) are satis ed, there exists one and only one sequence fwN g which satis es (1.4) and which is in the open interval I . s wn+s Y n+k = Moreover, fwN g is strictly decreasing and satis es (1.2). nlim !1 w n k=1

+ ak ) = 2(1 1 uniformly in s 2 Z+ , where k def 2 + bk > 1. For scale-irregular aaagaskets (i.e. bN = aN ), it also holds that I = (0; 1) and N !?1 lim wN = 1.

The ratio of the rate for a o-horizontal to horizontal jump of XN;wN is wN , hence (1.2) means that on small scales the process favors horizontal moves, while wN > 0 means that the process span the whole fractal space and is not con ned 3

in a line. For spatially symmetric scale-irregular aaa-gaskets, limN !?1 wN = 1, which implies that isotropy is asymptotically restored. Not much studies has been done on this diusion. The asymptotically one-dimensional diusion is rst constructed on the Sierpinski gasket [11]. [20] gives a characterization of the diusion on the Sierpinski gasket by exit distributions. The fractals may be regarded to have `obstacles' or holes in the space, when compared to uniform Euclidean spaces. Intuitively, a random walker that favors horizontal motion performs a one-dimensional random walk between a pair of obstacles, and eventually is forced to move in o-horizontal direction before they could move further horizontally. There are obstacles of various scales (sizes), separated by distances of the same order as their scales, hence globally, the random walker is scattered almost isotropically. This picture may be regarded as a new mechanism of homogenization, unique to fractals. The intuitive picture of isotropy restoration can be made mathematically persuasive through analyses of renormalization groups [11, 10]. The intuition implicitly guided the studies in [11], but in spite of the generality in the intuition, it was not clear how to obtain such diusions for fractals which lack exact self-similarity. Also the statements for the diusion in that work were not referring to the properties which explicitly embodied the picture. It is the purpose of this paper to report some positive answers (Theorem 1.2) to these points. We construct the diusion as a weak limit of XN;wN ([LN t]), N 2 Z, for a time-scaling constant LN . A key step in the construction is an asymptotic estimate of number of steps of XN;wN , whose expectation value is LN . We apply a limit theorem for discrete-time multi-type supercritical branching processes with singular and irregular environment. We need multi-type branching processes because the horizontal and o-horizontal jumps have dierent transition probabilities. Branching rates change with the generation N (irregular environment) because the substructure of a triangle in the pre-gasket varies with its scale N . Environment varies also because the transition probabilities of the random walk XN;wN vary with N . In particular, birth rates of types corresponding to the numbers of o-horizontal jumps approach 0 as N ! 1 (singular environment). Compared with existing related results, there are two major complications arising from these requirements; criterion for supercriticality, and scaling factor for total number of descendants. For a construction of spatially symmetric diusion on an exact self-similar nitely rami ed fractal [15, 3, 17, 14], the expectation of o-spring for the associated branching process is a constant matrix independent of generation N . For a construction of asymptotically one-dimensional diusion on an exact self-similar nitely rami ed fractal [11, 13], the o-spring expectation matrix has a limit as N ! 1. In these cases, the largest eigenvalue of the (limit of) o-spring expectation matrix gives the (asymptotic) growth rate of descendant expectations and governs supercriticality. A pioneering work for scale-irregular fractals by Hambly [7] deals with spatially symmetric diusions on fractals called HSG( ) (which have much 4

in common with scale-irregular aaa-gaskets as far as construction of diusions are concerned). Due to the spatial symmetry, the associated branching process is of one-type, hence the o-spring expectation is one-dimensional, which gives the growth rate. In the present study, o-spring expectation is a multidimensional matrix, and neither is constant nor has a limit. Thus a criterion for supercriticality cannot be given in terms of growth rates. Furthermore, the ratios of expectations of the population between dierent types are unbounded, which obscures at rst site, the existence of scaling factor for total descendant numbers. Our approach is partly inspired by a study on multi-type branching processes in random environment by Cohn. Much of our proof of Proposition 2.1 follows the idea in [5]. In that work, a probability measure on environments is considered, and the assumptions on stationarity and ergodicity implicitly assured the last two assumptions in Proposition 2.1 (including supercriticality) to hold almost surely. This approach is not suitable for our purpose to consider singular environment, where some of the branching rates vanish in the limit. To formulate a sucient condition of supercriticality in Theorem 2.2, we introduce a recursion relation in Appendix B which re ects recursive nature of branching processes. We apply this recursion relation also to prove continuity of limit distribution in Theorem 2.5. The idea of using recursion relation to prove continuity originally appeared in [11, Lemma 2.7], which we re ne to put it into a framework of branching processes, and generalize to handle irregular environments. It turns out in Proposition 2.4 that the existence of scaling factor for total descendant follows from a fact that the distribution of normalized population converge to a limit independent of types. Consideration on the branching processes may be interesting in its own respect, so we will discuss this in Section 2 independently of other sections. To apply the general theory of branching processes to the diusion, we consider in Section 3 estimates for generating functions. The calculations are lengthy, for some of which the author used REDUCE, a computer program for rigorous formula reduction (Proposition 3.1). We use these basic estimates to obtain estimates for number of steps of XN;wN , to which one can apply [11, Sect. 3].

2 Branching process with singular and irregular environment.

Let d  2 be an integer and put E def = f1; 2; 3;


; dg. Consider a discrete time d-type branching process Z~ N = (ZN;j , j 2 E ), N 2 Z+ . Given n 2 Z+ and i 2 E , let = (Zn;N;i;j ; j 2 E ); N = n; n + 1;


; Z~ n;N;i def 5

be random vectors which give the distribution of the number of descendant at time N from a single ancestor of type i at time n. We have, for r 2 Z+ and j 2 E,

Zn+r;j =

n;i X ZX

i2E u=1

Zn;n+r;i;j;u ;

where (Zn;n+r;i;j;u ; j 2 E ), u 2 Z+ , are i.i.d. copies of Z~ n;n+r;i when conditioned on Zn;i . Let feng be a non-negative sequence. Proposition 2.1. Assume that following three conditions hold for each i 2 E . = E[ZZnNij ] ; (1) Uniform estimates for second moments of WnNij def nNij 2 ] < 1; j 2 E ; v def = sup sup en E[ WnNij

n2Z+ N n+n0 2 ; WnNij lim sup E[ WnNij p!1 N n+n0

> p ] = 0; j 2 E ; n 2 Z+ ;

for some constant n0 2 Z+ .

E[ ZnNij ] > 0 exist, positive and indepen= Nlim (2) For each n 2 Z+ , ni def !1 E[ ZnN 1j ] dent of j 2 E . (3) eN ZNj diverges in probability to in nity; Nlim Prob[ eN ZNj  p ] = 1, !1 j 2 E , p > 0. Then the sequence of normalized random vectors (ZNj =E[ ZNj ]; j 2 E ) converges in L2 as N ! 1 to a random vector (W; W;


; W ) with E[ W ] = 1. Proofs of the results in this section is postponed to the end of the section. Generalization to include eN is for our application in Section 3. A simple sucient condition for the existence of ni is given in Appendix C, in terms of o-spring expectation matrices AN (ANij = E[ ZN ?1;N;i;j ]). The last assumption states supercriticality. One of our main concern here is a useful condition for the last assumption to hold. De nition. A family of sequences of pairs of reals f(xk;n ; yk;n), n = 0; 1;


; kg, k 2 Z+ , is said to satisfy the assumption R, if there exist sequences of nonnegative numbers fang, fwn g, fwn0 g, n 2 Z+ , satisfying 2  inf n an , supn an < 1, and maxfwn ; wn0 g  minf1; Cw ?n g, n 2 Z+ , for constants Cw > 0 and  > 1, such that an+1 ; y an+1 + w xk;n  xk;n n+1 minf1 ? xk;n +1 k;n+1 g ; +1 0 yk;n  xk;n+1 + wn+1 yk;n+1 ; 0  n  k ;

6

hold for all k 2 Z+ . Similarly, f(xk;n ; yk;n ), n 2 Z+ g, k 2 Z+ , is said to satisfy the assumption R, if similar relation hold for n 2 Z+ and k 2 Z+ .

Theorem 2.2. Assume that for some j0 2 E , Z0;j = 1, j = j0, and Z0;j = 0, otherwise. Let p > 0 and j 2 E . Suppose that there exists an integer n0 and a non-empty subset E 0  E , not equal to E , such that the family of sequences f(xk;n ; yk;n )g de ned by xk;n = max Prob[ en0 +k Zn;n0 +k;i;j < p ] ; i2E 0 yk;n = i2EnE max0 Prob[ en0 +k Zn;n0 +k;i;j < p ] ; 0  n  k; k 2 Z+ ; satis es the assumption R, and

(2.1) Then

lim inf

k!1; xk;k 6=0

(

(? log xk;k )

Yk )1=k =0

a

> 1:

lim Prob[ eN ZN;j  p ] = 1 :

N !1

The assumption (2.1) is an `a priori estimate' that Prob[ en0 +k Zk;n0 +k;i;j < p ] is not too large. The Theorem then says that it is in fact small. Let us call the types j 2 E 0 the dominant types, and the types j 62 E 0 the recessive types. The assumption R re ects the recursive nature of branching process. It is satis ed when the probability that recessive types appear in the o-springs of a parent at generation n vanishes exponentially as n ! 1, and if recessive type do not appear in the o-springs, then at least one dominant type o-spring appears from a recessive parent, while at least an+1 (no less than 2 but bounded) dominant o-springs appear from a dominant parent. fwn g represents singular environment, while n-dependence of fang implies irregular environment. Though the branching rate to recessive types vanish in the limit, the recessive types may contribute signi cantly to the growth of dominant types, because a recessive parent may give birth to exponentially many dominant type o-springs. Thus in general, we can not discard the recessive types from consideration for limit theorems. Note also that supercriticality is non-trivial because of the recessive types. Assumption on initial condition Z0;j is chosen to be simple, to avoid complications. The following is useful in obtaining an a priori estimate of type (2.1) from moments of ZnNij . Proposition 2.3. Let p 2 R pand X a real valued random variable. If E[ X ] > p then Prob[ X  p ]  1 ? d + d2 ? 1, where d = 1+2?1 V[ X ] (E[ X ] ? p)?2 . 7

The next statement is on the existence of norming factor for total descendant numbers. Proposition 2.4. Assume that the sequence (ZN;j =E[ ZN;j ], j 2 E ), N 2 Z+ , converges P Z in probability as N ! 1 to a random vector (W; W;


; W ). Then P j2EE[ ZN;j ] converges in probability as N ! 1 to W . j 2E

N;j

We complete our list of the results with a sucient condition for the continuity of the limit distribution, stated in terms of the assumption R. Let Wn;i , n 2 Z+ , i 2 E , be real valued random vectors, and let




?p

n;i (t) def = E[ exp ?1 t Wn;i ] ; denote the characteristic function. We assume an `a priori' estimate of the form (2.2) jn;i (t)j  1 ? Cn t2 ; ?tn < t < tn ; n  n0 ; i 2 E 0 ; for some non-empty subset E 0  E not equal to E , an integer n0 , and positive reals Cn and tn . Theorem 2.5. Assume that ftk ; k 2 Z+g in (2.2) diverges to in nity as k ! 1 exponentially fast at most (lim sup t1k=k < 1), and satis es k!1

tj > 0 :  def = k; 9inf sup tj 0 ; i 2 E; j 2 E ;

(3.6) A~(a; b; 0)Ar;Ar  (a + 1)2 : For each i 2 E put ga;b;i (w; h) = F~i (a; b; w; ~ (w)+ w h). Then ga;b;i is a rational function in h 2 CE and w, analytic at h = 0 for w 2 I , and for each j1 ;


; j4 2 E,

1 @ 2g a;b;j 1 (w; h = 0) < 1 ; sup w2I w @hj2 @hj3

and

1 @ 3g a;b;j 1 (w; h = 0) < 1 : sup w2I w @hj2 @hj3 @hj4

Proof. A(a; b; w) has non-negative elements because it is an expectation matrix for number of steps. From graphical considerations we see that every type j of steps appear with positive probability for any i, hence they are positive. F~ is related to the generating function for number of steps of random walks (see also Proposition 3.2 below), from which we see that ga;b;i (w; h) are rational functions both in h and w, and analytic at h = 0. The parameter w is the relative jump rate of the random walk. Therefore for w 2 I there are no singularities. The only possible relevant singularities of F~ are at w = 0. The estimates in the statement are proved by explicit calculation of F~ with aid of computer. We give in Appendix D explicit form of A~(a; b; w = 0) obtained as the rst derivatives of F~ using REDUCE. The explicit formula implies (3.4), (3.5), and (3.6). The estimates on higher derivatives of F~ at w = 0 is also obtained using REDUCE. See Appendix D for more information on computer aided proof of this Proposition. 2

Note that (3.4) and (3.5) imply (3.7)

inf w2I

X~

k2E

A(a; b; w)kj > 0; j 2 E :

We go back to the gasket and look into the N -dependence. Assume that  def = f(aN ; bN ); N 2 Zg is a bounded sequence of pairs of integers satisfying (1.1), which determines the gasket G. For N 2 Z, let FN = (FN;Ap ;


; FN;Er ) be a CE -valued function in ]E (= 8) variables de ned by (3.8)

FN;i (u) def = Fi (aN ; bN ; u) ; i 2 E : 14

= diag((wN )). Using FN and N , we can Also de ne a diagonal matrix, N def write the generating functions for the number of steps of XN;wN . Let i 2 E , j 2 E , and n  N . Let ZnNij be the random variable which counts the number of steps of type j 2 E between the times Tn;1(XN;wN ) and Tn;0 (XN;wN ), under the condition that (XN;wN (Tn;0 (XN;wN )); XN;wN (Tn;1 (XN;wN ))) forms an edge of type i in Hn . Tn;i is a hitting time of Gn de ned in Section 1. By strong Markov property of simple random walks, the distribution of ZnNij is independent of the starting point of XN;wN , and the random variables which count the number of steps of type j 2 E between the times Tn;k+1 (X ) and Tn;k (X ), k = 0; 1; 2;


, are independent and equal in distribution to ZnNij , under similar conditions. By de nition, Znnij = 1 (j = i); = 0 (j 6= i). Proposition 3.2. Fix n 2 Z and i 2 E . (ZnNij ; j 2 E ), N = n; n + 1;


, is a multi-type branching process whoseYgenerating functions nN = (nN;Ap ;


; nN;Er ) de ned by nNi (z ) def = E[ zjZnNij ], satisfy, for n < N , j 2E

(3.9)

nN (z ) = ?n 1 (Fn+1 


 FN )(N z ) ; z 2 CE :

Proof. The strong Markov property of simple random walks and the nite rami edness of the fractal imply that fZnNij g is a branching process. In particular,

(3.10)

nN (z ) = n;N ?1 (N ?1;N (z )) ; N > n;

holds. The de nitions of N and FN imply (3.11)

n;n+1 (z ) = ?n 1 Fn+1 (n+1 z ) ;

2

which, together with (3.10) implies (3.9).

Proposition 1.3 implies that some o-spring branching rates vanish as N !

1, hence we are considering branching process with singular environment. For integers n and N satisfying n  N , de ne ]E -dimensional matrices

= An+1 An+2


AN : = A~(aN ; bN ; wN ); B~nN def = diag(~ (wN )); A~N def ~ N def Then (3.9), (3.1), and (1.4) imply (3.12)



E[ ZnNij ] = ~ ?n 1 B~nN ~ N



ij

; i 2 E; j 2 E; N  n :

Elementwise positivity of AN and hence of BnN were noted in Proposition 3.1; (3.13) A~Nij > 0; B~nNij > 0; i; j 2 E ; N > n : 15

Proposition 3.3. (1) For each n0 2 Z there exist positive constants C1 and C2 such that if n 2 Z and N 2 Z satisfy N ? 2  n  n0 , then B~nNij  C1 2 B~nNij wwN2 n

 C2

(2) There exist limits

N Y

(ak + 1)2 ;

k=n+1 N  Y

1 + bk

k=n+1

2

2

; i 2 E; j 2 E:

~

BnNij ; n 2 Z; i 2 E ; j 2 E ; = Nlim

ni def !1 B~nN 1j independent of j , which satisfy, for each n0 2 Z, 0 < nninf;i2E ni  sup ni < 1 : nn0 ;i2E

0

(3) For each n0 2 Z there exists a positive constant C3 such that if integers n, m, and N satisfy m ? 1  n  n0 and N  maxfm; n + 2g, then X~ X~ Bn;m?1;i;k BmNk0 j  C3 B~nNij ; i 2 E ; j 2 E : k0 2E

k2E

Proof. Since F is rational in w, A~ is also rational. Therefore (3.6) implies A~k22  (ak + 1)2 + CP4 wk , k 2 ZP , where C4 is a positive constant. Proposition 1.3 implies that kn wk < kn0 wk < 1, n  n0 , hence we obtain the rst estimate for B~nNij . Proposition 1.3 implies that for each n0 there exists a N Y constant C5 > 0 such that wwN  C5 k?1 , N  n  n0 . With the rst n k=n+1 estimate, we have the second estimate. The estimates (3.4) and (3.5) imply that the elementwise positive matrices A~N , N 2 Z, satisfy the assumption of Theorem C.1 in Appendix C with q = 2. Theorem C.1 then implies the second assertion.  def = f(aN ; bN ); N 2 Zg is a bounded sequence, hence contains nite number of distinct pairs; as far as  is concerned, taking supremum or in mum in N is taking maximum or minimum among nite possibilities. Assume N  m + 2. Then

B~nNij 



X~ k

X~

k2E

Bn;m?1;i;k

Bn;m?1;i;k

X~ k0

X

k0 2E

~ A~mkk0 kmin 00 2E BmNk00 j

BmNk0 j `2E ;m inf0n

X~

0 `0

16

Am0 ``0

inf fB~ 00 =B~ g k00 mNk j mN 1j

]E supfB~mNk00 j =B~mN 1j g k00

:

It is now easy to see that the second assertion and (3.7) imply the third assertion. The cases N = m and N = m + 1 can be proved similarly. 2 = E[ZZnNij ] . Put WnNij def nNij Proposition 3.4. Let i 2 E and j 2 E . For each n0 2 Z there exists a positive constant C such that for all N and n  n0 satisfying N  n + 2, 2 ]C ~ ni wn?1 : E[ WnNij 3 ] < 1. Also, for each n 2 Z the third moment is bounded in N ; sup E[ WnNij

N n

Proof. By taking derivatives of nN in Proposition 3.2 we obtain 2 ] E[ WnNij N   @ 2F~k1 X X ~ ?n 1 B~n;m?1 (am ; bm ; wm ; ~ (wm )) = f k1 ;k2 ;k3 2E m=n+1

~

 ~

i;k1



@uk2 @uk3

 ~ ?1 ~



+ n Bn;N ~ N g E[ ZnNij ]?2 : B ~ i;j k2 ;j m;N N k3 ;j Proposition 1.3 implies that for each n0 there exists a constant C > 0 such that

 Bm;N ~ N

wN  C w n

N Y

k=n+1

k?1 ; N  n  n0 :

With Proposition 3.1, (3.12), and Proposition 3.3 we have the bound for the second moment. The bound on the third moment is proved in a similar way. 2 Let E 0  E be the set of horizontal edges; (3.14) = fAr; Br; Erg : E 0 def Note that Table 1 and (3.2) imply that there exist positive constants C and C 0 such that for w 2 I we have (3.15) C  ~ i (w)  C 0 ; i 2 E 0 ; C w  ~ i (w)  C 0 w ; i 2 E n E 0 : De ne, for N 2 Z+ , XX = (3.16) LN def E[ Z0Nij ] :

i2E j 2E The next Theorem shows that LN is the appropriate scaling factor Xfor the random walk X = XN;WN on HN . Note that Tn;1 (X ) ? Tn;0(X ) = ZnNij , j 2E

where i is the type of edge formed by the endpoints of X in this time interval. 17

Theorem 3.5. Let m 2 Z and i 2 E . (WmNij , j 2 E ), N = m; m + 1;


, converges in L2 (hence in probability and in law) as N ! 1. The limit is a random vector with equal components X (Wmi ;


; Wmi ), satisfying E[ Wmi ] = 1 and sup wn E[ Wni2 ] < 1. L?N1 ZmNij converges in probability as N ! 1 to nm 0 def Wmi = P

j 2E

mi Wmi , with mi as in Proposition 3.3. The distribution E[ j;k Z0mjk ] mk

0 is continuous. of Wmi

Proof. As noted in Proposition 3.2, (ZmNij , j 2 E ), N = m; m + 1;


, is a branching process. The number of descendant at time N from a single ancestor of type k at time n is equal in distribution to that of (ZnNkj , j 2 E ). Fix j 2 E . Proposition 3.4 implies, with (3.15), 2 ] < 1: sup sup wn E[ WnNij

nm N n+2

Furthermore, with E[ WnNij ] = 1 and the Schwarz inequality, this implies the uniform integrability of WnNij ; plim E[ WnNij ; WnNij > p ] = 0. This !1 Nsup n+2 3 ] in Proposition 3.4 and the Schwarz with the uniform bound for E[ WnNij inequality implies 2 ; WnNij > p ] = 0 ; n  m : lim sup E[ WnNij

p!1 N n+2

E[ ZnNij ] exists, Proposition 3.3 with (3.12) implies that for each n  m, Nlim !1 E[ ZnN 1j ] positive and independent of j 2 E . Hence, if we prove that wN ZmNij diverges in probability to in nity, then all the assumptions of Proposition 2.1 will be satis ed, with eN replaced by wm+N and n0 = 2. Proposition 2.1 then will imply that (WmNij ; j 2 E ) converges in L2 as N ! 1, to a random vector with equal components. By de nition, Zmmij = 1 (j = i) and = 0 (j 6= i). Fix p > 0 and j 2 E , and de ne a family of sequences f(xk;n ; yk;n )g 0  n  k, k 2 Z+ , by

xk;n = max Prob[ wm+n0 (k)+k Zm+n;m+n0(k)+k;i0 ;j  p ] ; i0 2E 0 yk;n = i0max Prob[ wm+n0 (k)+k Zm+n;m+n0(k)+k;i0 ;j  p ] ; 2EnE 0 where E 0 is de ned in (3.14). n0 (k) is an arbitrary function of k taking nonnegative integer values (to be speci ed later). De ne a sequence fa~n ; n 2 Z+ g by a~n = am+n + 1, and fw~n ; n 2 Z+ g by = max w~n def Prob[ i0 2E 0

X

k2EnE 0

Zm+n?1;m+n;i0 ;k  1 ] :

18

w~n is the largest probability among i0 2 E 0 that the random walk X = XN;wN with N = m + n jumps o-horizontally at least once in the time interval [TN ?1;1(X ), TN ?1;0(X )], under the condition that the endpoints of X for this time interval forms an edge of type i0 in HN ?1 . By de nition, 3  inf n a~n  supn a~n < 1. Also 0  w~n  1 and is of order wm+n , for which Proposition 1.3 implies w~n  C1 ?n for some constant C1 > 0 and  = min  = k k 2(1 + ak ) > 1 (recall that there are only nite number of distinct pairs min k 2 + bk (ak ; bk )). From graphical consideration we see that xk;n

8 < X a~n+1  max Zm+n;m+n+1;i0 ;j0 = 0 ] xk;n +1 0i 2E 0 :Prob[ j 0 2EnE 0 9 = X +Prob[ Zm+n;m+n+1;i0 ;j0  1 ] yk;n+1 ; : j 0 2EnE 0 X

We may either use yk;n  1 or use Prob[

j 0 2EnE 0

Zm+n;m+n+1;i0 ;j0 = 0 ]  1, to

conclude that xk;n satis es the inequality in the de nition of the assumption R in Section 2, with fwn g and fan g replaced by fw~n g and fa~n g, respectively. Similarly, we nd

yk;n

8 < X  i0max Prob[ Zm+n;m+n+1;i0 ;j0  1 ] xk;n+1 2EnE 0 : j 0 2E 0 9 = X +Prob[ Zm+n;m+n+1;i0 ;j0 = 0 ] yk;n+1 ; ; j 0 2E 0

from which we know that yk;n also satis es the inequality of the assumption R. For k 2 Z+ and i0 2 E 0 put

= E[ wm+n0 (k)+k Zm+k;m+n0 (k)+k;i0 ;j ] ; ek;i0 def vk;i0 def = V[ wm+n0 (k)+k Zm+k;m+n0 (k)+k;i0 ;j ] : Proposition 3.3, (3.12), (3.15), and bk  2 imply ek;i0  C2 wm2 +k 4n0 (k) , where C2 > 0 is a constant independent of n0 2 Z+ and k 2 Z+ . For each k, de ne n0 (k) to be suciently large so that ek;i0  2p for all k 2 Z+ . Proposition 2.3 then implies that Prob[ wm+n0 (k)+k Zm+k;m+n0 (k)+k;i;j  p ] < 1 ? 1=(2d), where d  1 + 2 vk;i0 e?k;i20 . Applying Proposition 3.4 and (3.15) we see that xk;k < 1 ? C3 wm+k , where C3 is a positive constant independent of k. Proposition 1.3 implies wm+k  C4 wm

Yk

=0

m?1+ with k = 2(1 + ak )=(2 + bk ), and 19

for a positive constant C4 independent of k. Hence (? log xk;k )

Yk

=0

a~ > C3 C4 wm

Yk

(1 + bm+ =2) ;

=0 ( Yk )1=k which implies k!1 lim; xinf 6=0 (? log xk;k ) a~ k;k =0

 2 > 1. We see that all

the assumptions in Theorem 2.2 are satis ed, with eN replaced by wm+N , and ZN;j by Zm;m+N;i;j . Theorem 2.2 then implies Nlim Prob[ wN ZmNij  p ] = !1 1 ; p > 0, which, as we noted in the rst part of the proof, proves the convergence of (WmNij ; j 2 E ) to (Wmi ;


; Wmi ). Weak convergence and uniform

integrability imply convergence in expectations. Therefore, from what we have 2 ]. proved, we obtain the statements on E[ Wmi ] and E[ Wmi Let n 2 Z+ . The j independence of ni in Proposition 3.3 implies, with (3.12), P E[ Z ] nNij j P lim N !1 E[ Z ] = ni : nN 1j

j

Also from (3.12) one sees, for m  n  N , E[ ZmNij ] =

X

k2E

E[ Zmnik ] E[ ZnNkj ] :

X

E[ ZnNij ] = P E[ Zni ] . j;k P0njk nk j Z Convergence of (WnNij ; j 2 E ) and Proposition 2.4 imply that P E[j ZnNij ] nNij j converges in probability to W . Therefore we have the convergence in probni X ? 1 0 ability of LN ZnNij to Wni . With E[ Wni ] = 1 and Proposition 3.3, we Hence, with (3.16), we see that Nlim L?1 !1 N

have (3.17)

j 2E

X

E[ Wni0 ] = P E[ Zni ]  C5 ( E[ Z0njk ])?1 ; 0njk nk j;k j;k for some positive constant C5 independent of n  0 and i 2 E . Similarly, there exists a positive constant C6 such that

(3.18) Let

E[ Wni0 2 ]  C6 (

X j;k

E[ Z0njk ])?2 wn?1 ; n  0; i 2 E :

?p




0 ]; t 2 R; n;i (t) def = E[ exp ?1 t Wn;i denote the characteristic function. With obvious bound 0  1 ? 1;

is satis ed with a replaced by an0 ++1 + 1. Proposition 3.3 and (3.12) imply klim t = 1, while boundedness of A~nij !1 k implied in (3.4) with Proposition 1.3 and (3.15) gives supk0 t1k=k < 1. Let n  0 and m  0. From (3.12), (3.15), Proposition 3.3, Proposition 1.3, and bk  2, we see

CP8 wn tn < ?m tn+m wn+m min` k2E 0 Bn;n+m;`;k < C8 4 ; where C8 is a positive constant independent of n and m. Therefore there exists an m0 such that t tn < 1 for all n  0. With the boundedness of Anij we also n+m0 t n > 0. Hence all the assumptions for ftk g in Theorem 2.5 hold. see ninf 0 tn+m0

Using [11, (2.30)], we can proceed with similar arguments as we did for Prob[ wm+n0 (k)+k Zm+n;m+n0(k)+k;i0 ;j  p ], from which we see that n;i satis es the assumption R condition of Theorem 2.5. We have now proved that n;i satis es all the assumption of Theorem 2.5, which implies that the distribution 0 is continuous. of Wn;i 2 Let D def = D([0; 1); G) be the set of cadlag paths on the scale-irregular abbgasket G. For n 2 Z and x 2 Gn we de ne a family of probability measures Px(N ) [
], N = n; n + 1;


, on D, by Px(N )[ w(0) = x ] = 1 and

Px(N )[ w(ti ) = xi ; i = 1; 2;


; r ] = Prob[ XN;wN ;x ([LN ti ]) = xi ; i = 1;


; r ]; 21

where XN;wN ;x is the random walk XN;wN with starting point x; XN;wN ;x = x. We use abbreviations such as

Px(N ) [ w(0) = x ] def = Px(N ) [ fw 2 D j w(0) = xg ] ; and write Ex(N ) [
] for the expectations with respect to Px(N ) [
]. De ne = Tn;i (w), w 2 D, similarly as we did in Section 1 for processes, and put Wn;i def Tn;i+1 ? Tn;i . Let N  n, x 2 GN , i 2 Z+ , and let x0 ; x1 ;


; xi be a sequence of points in Gn such that each adjoining pair is an n-neighbor pair and x0 and x are in a unit triangle of Hn . Consider the distribution of Wn;j , j = 0; 1;


; i ? 1, under the conditional probability Px(N )[
j w(Tn;j (w)) = xj ; j = 0;


; i ]. Since the probability is based on random walks, this distribution is a direct product of the distributions of each Wn;j , and as we noted before Theorem 3.5, the distribution P of each Wn;j under the conditional probability is equal to that of L?N1 `2E ZnNk` , if (xj ; xj+1 ) forms an edge of type k 2 E , and is independent N ) [
], and their of i, j , x, and xj 's. We denote this distribution of Wn;j by Q(n;k limit distributions as N ! 1 by Qn;k [
], k 2 E . Theorem 3.5 implies (3.19) Nlim Q(N )[ s j a < s < b ] = Qn;k [ s j a < s < b ] ; 0  a < b  1 : !1 n;k We need a following type of uniformity to handle processes starting from `irrational' points. Proposition 3.6. Let N , M , n be non-negative integers satisfying N  M  n, and let x 2 GM and y 2 Gn such that x and y are in a unit triangle of Hn . Then there exists a positive constant C9 independent of x, y, n, M , and M , such that

Ex(N ) [ Tn;0 (w) j w(Tn;0 (w)) = y ]  C9

n  Y

`=1

1 + b2`

?2

:

Proof. By similar arguments for the proof of [11, (3.2)], we see that there exist positive constants C10 and C11 such that for Xm = Xm;wm;x0

(3.20)

E[ Tm?1;0 (Xm ) j Xm (Tm?1;0 (Xm )) = y0 ]  C9 + Cw10 ; m

for all m 2 Z+ , x0 2 Gm nGm?1 , and y0 2 Gm?1 , with x0 and y0 in a unit triangle of Hm?1 . For an m-neighbor pair (u; v) forming a type k edge, Proposition 3.3, (3.16), and (3.12) imply (Tm;i ) = u; XN;wN (Tm;i+1 ) = v ] LN?1 E[ X Wm;i (XN;wN ) j XN;wNX (3.21) = L?N1 ZnNk`  C11 ~ ?nk1 ( B~0:n?1;`;`0 )?1 ; `2E

`;`0

22

where C11 is a positive constant independent of m, N , u and v. Proposition 3.3, (3.20), and (3.21) imply (3.22) Ex(N0 ) [ Tm?1;0 (w) j w(Tm?1;0 (w)) = y0 ]  C12

m Y

`=1

(1 + b` =2)?2 ;

for all N  m  0, x0 2 Gm n Gm?1 , and y0 2 Gm?1 , with x0 and y0 in a unit triangle of Hm?1 . C12 is a constant. The estimate (3.22), combined with the strong Markov property of the random walks, implies for N  M  n, x 2 GM , y 2 Gn , with x and y in a unit triangle of Hn ,

Ex(N ) [ Tn;0 (w) j w(Tn;0 (w)) = y ] M XX Ey(Ni ) [ Ti?1;0 (w) j w(Ti?1;0 (w)) = yi?1 ] = fyi g i=n+1

 Px(Nn)[ w(Ti;0 (w)) = yi ; n + 1  i  M ? 1 j w(Tn;0 ) = y ] Y  C9 (1 + b` =2)?2 ; `=1

where the rst summation is taken over fyig = (yn ; yn+1 ;


; yM ) with yi 2 Gi , yn = y, yM = x, such that yi and yi?1 are in a unit triangle of Hi?1 , for i = n + 1;


; M . 2 The following result is used to prove that an N -decimated walk of a diusion, obtained as the continuum limit N ! 1 of a sequence of random walks, is equal to the original random walk. Proposition 3.7. For N 2 Z+ , let XN be a simple random walk on GN with N -neighbor jumps. Assume that there exists a sequence LN diverging to in nity as N ! 1 such that, X~N (
) def = XN ([LN
]) converges almost surely as N ! 1 to some continuous strong Markov process X (
) on G. Let n 2 Z+ . If for each N  n, the n-decimated walk (de ned in Section 1) of XN is equal in law to Xn , then the n-decimated walk of X is also equal in law to Xn . Proof. Fix x 2 Gn and y 2 Gn . Denote by P (x) [
] the conditional probability with condition XN (0) = x, N 2 Z+ , X (0) = x, and let E (x) [
] be expectation with respect to P (x)[
]. For N  n and a positive integer q, de ne N;q def = def ~ inf ft  0 j d(XN (t); Gn n fxg)  1=qg, q = inf ft  0 j d(X (t); Gn n fxg)  = inf ft  0 j X (t) 2 = inf ft  0 j X~N (t) 2 Gn n fxgg, 1 def 1=qg, N;1 def Gn n fxgg, where d is the metric on G. The almost sure convergence of X~N to X implies (3.23)

q  lim inf   lim sup N;q  q+1 ; a:s:; q > 0 : N !1 N;q N !1

23

De ne a harmonic function h : G ! [0; 1] as follows; for z 2 G1 , i.e., z 2 Gm for some m  n, de ne h(z ) def = Prob[ X~m (m;1 ) = y j X~m (0) = z ]. The assumption on the decimation property implies that Prob[ X~ m0 (m0 ;1 ) = y j X~m0 (0) = z ] is constant for m0  m, hence h is well-de ned on G1 . We can see that [11, Proposition 3.2] holds in our case, which implies that h is continuous. In particular, h is uniquely extendable as continuous function to G. By de nition, h(y) = 1 and h(y0 ) = 0, y0 2 Gn n fyg. XN is a simple random walk, and h, restricted on GN , is an associated harmonic function. Therefore h(X~N (t ^ N;q )), t  0, is a martingale (a ^ b def = minfa; bg), hence E (x) [ h(X~N (t ^ N;q )) ] = h(x), N  n. This with (3.23), Nlim X~ = X , and !1 N continuity of h implies

E (x) [  min h(X (t ^ s)) ]  h(x)  E (x) [  max h(X (t ^ s)) ] : q sq+1 q sq+1 (x) Continuity of X implies that qlim !1 q = 1 . Hence we have h(x) = E [ h(X (t^ 1 )) ], t  0. Since this is independent of t, we have h(x) = E (x)[ h(X (1 )) ] = Prob[ X (1 ) = y j X (0) = x ], which implies that the transition probability of n-decimated walk of X is equal to that of Xn (0). 2

Proof of Theorem 1.2. We can apply [11, Sect. 3], with [11, Theorems 2.5, 2.8] replaced by Theorem 3.5, [11, (3.1)] by (3.19), and [11, Proposition 3.1(1)] by Proposition 3.6. Then for xN 2 GN , N 2 Z+ , satisfying Nlim x = x, the !1 N ( N ) sequence of measures PxN [
] (the distribution of XN;wN ;xN ([LN t])), N 2 Z, converges weakly as N ! 1 to a symmetric Feller process X . Skorokhod's Theorem implies that there exists a probability space and GN valued processes XN , N 2 Z, such that XN is equal in law to XN;wN ;xN and converges almost surely to a process equal in law to X . Proposition 3.7 implies that the ndecimated walk of this process is equal in law to the original random walk Xn;wn . That this random walk has the asymptotically one-dimensional (and isotropy restoration) properties, is proved in Proposition 1.3. 2 Acknowledgements. The author would like to thank Prof. K. Hattori and Prof. H. Watanabe for collaborations on which the present work stands, and Prof. S. Kusuoka for the proof of Proposition 3.7. He would also like to thank Prof. M. T. Barlow and Prof. H. Tasaki for valuable discussions concerning the present work, Prof. C. Burdzy for the hospitality and discussions during the author's stay at University of Washington, Prof. T. Nakagawa for bringing references on branching processes to the author's attention, and Prof. B. M. Hambly, Prof.

24

H. Osada, Prof. T. Kumagai, and Prof. Z. Vondracek, for kindly sending related works before publication.

Appendix A Metric for scale-irregular abb-gaskets. We de ne a metric d for a scale-irregular abb-gasket. Consider a[sequence of vertices of pre-gaskets, G0  G1  G2 


, and let G1 = Gn , as in Section 1. Let x 2 G1 and y 2 G1 , i.e. x 2 GN , y 2 GN , for some N . We explain the case where x and y are in a unit triangle of G0 . The case where x and y are `further apart' can be handled similarly. Let path(x; y) be a collection of nite sequences z = fz0 = x; z1 ;


; zK = yg for some K = K (z )  0, which has a property that for each i = 0; 1;


; K (z )?1, there exists sz (i) 2 Z+ such that zi and zi+1 are sz (i)-neighbor pairs. Let faN ; bN g be a bounded sequence of pair of positive integers, that determines n Y G1 . Put n = minfak +1; bk +1g?1, n 2 Z+ . For z 2 path(x; y) put L(z ) = KX (z)?1

k=1

sz (i) . Then the metric d is de ned to be d(x; y) def = z2path inf(x;y) L(z ). It is i=0 straightforward to see that if x and y are an N -neighbor pair, then (A.1) d(x; y) = N : That d is a metric is also easy to see. Yn ?1 The de nition of n can be replaced by n = `k , for any set of `k k=1 satisfying `k  minfak + 1; bk + 1g and inf k `k > 1. The rst condition implies (A.1). The second condition with the boundedness of faN ; bN g implies that there exists a positive constant C such that if x and y is in a unit triangle of GN then d(x; y)  C N . This property means that our de nition of d is

compatible with our analysis in this paper based on the hierarchical (recursive) structure of the fractal.

B Decay estimates from non-linear recursion relations. The Lemma below gives a mild decay estimate from a non-linear recursion relation. We apply the Lemma to prove a Theorem which states a sharp decay 25

estimate from another recursion relation with more involved assumptions. Lemma B.1. Let fwn; n 2 Z+ g be a sequence in [0; 1] satisfying Pn wn < 1, and fan; n 2 Z+ g a sequence satisfying D def = inf n an > 1 and supn an < 1. For each k 2 Z+ de ne a sequence fxk;n ; n = k; k ? 1;


; 0g by a recursion relation an+1 + w ; n = k ? 1; k ? 2;


; 0 ; xk;n = (1 ? wn+1 ) xk;n n+1 +1

with initial condition xk;k satisfying 0  xk;k  1. If

Yk

lim (? log xk;k ) a = 1 k!1; xk;k 6=0 =0 holds, then there exist positive constants C1 and k1 (independent of n and k) such that




?

xk;n  C1 sup w + exp ?Dfk ?n?1 ; 0  n  fk ; k  k1 ; n

where

p

fk = supfn  k ? 1 j D (? log xk;k )

Yk =n+2

a > 1g + 1 ;

Q with a convention k=k+1 a = 1, and fk = k if xk;k = 0. P Proof. Put C2 = supn exp(aY n ). n wn < 1 and p ?D1 > 1 imply that there exists a constant n1 such that (1 ? C2 w )  D , n  n1 . By assumption, n

lim f = 1, hence there exists a constant k1 such that fk  n1 ? 1, k  k1 . k!1 k Let k  k1 in the following. We rst prove that




?

xk;fk  exp ?D?1 ; k  k1 :

(B.1)

If xk;k = 0 then (B.1) directly follows, so we assume xk;k > 0. Put uk;n = ? log xk;n . The assumptions and the recursion relation imply 0 < xk;n  1 for all n, hence uk;n exist and are non-negative. Furthermore,





?an+1 ? 1) w uk;n = an+1 uk;n+1 ? log 1 + (xk;n n+1  an+1 uk;n+1 ; +1

which implies (B.2)

uk;n  uk;k

!

Yk =n+1

a ; 0  n  k :

26

The de nition of fk implies fk  k and the following three inequalities;

p

?1

uk;k  D ; if fk = k ;

(B.3)

p

(B.4)

p

(B.5)

D uk;k

0 k 1 Y A D uk;k @ a > 1 ; =fk +1 Yk ! =n+2

a  1 ; f k  n  k ? 1 :

From (B.2), (B.5), and D > 1 follows uk;n+1 < 1, fk  n  k ? 1, which, together with the recursion relation implies uk;n = an+1 uk;n+1 ? log(1 + exp(uk;n+1 an+1 ) wn+1 (1 ? exp(?uk;n+1 an+1 )))  an+1 uk;n+1 ? log (1 + C2 wn+1 uk;n+1 an+1 )  an+1 (1 ? C2 wn+1 ) uk;n+1 ; fk  n  k ? 1 :

Q

Q

This with k  k1 and (B.4) implies uk;fk  uk;k ( k=fk +1 a )( k=fk +1 (1 ? C2 w ))  D?1 , if fk  k ? 1. If fk  k then fk = k, hence (B.3) implies uk;fk > D?1 . Therefore we have (B.1). Put vn = sup w . fvn g is decreasing, bounded above by 1, and nlim !1 vn = 0. n D + vn+1 , 0  n  De ne a sequence fzn ; n = fk ; fk ? 1;


; 0g by zk;n = zk;n +1 ? 1 fk ? 1, and zk;fk = exp(?D ). Then

xk;n  zk;n ;

(B.6)

0  n  fk :

Put ?
(B.7) zk;n = exp ?Dfk ?n?1 + vn rk;n : Taylor's Theorem implies, with D > 1,



? ? rk;n  vnv+1 rk;n+1 D exp ?Dfk ?n?2 + vn+1 rk;n+1 D?1 + vn+1 ; n

which further is reduced with the properties of vn to

(B.8) rk;n  rk;n+1 D (e?1 + vn+1 rk;n+1 )D?1 + 1 ; 0  n  fk ? 2 : Put  = 21 (1 + D e?D+1). D > 1 implies 0 < D e?D+1 <  < 1. Therefore there exists a constant k2 de ned by

k2 = inf fn  0 j D (e?1 + vn (1 ? )?1 )D?1 < g : Monotonicity of fvn g implies D (e?1 + vn (1 ? )?1 )D?1 < , n  k2 . If fk  k2 +1 then we can prove by induction that rk;n  (1?)?1 , k2  n  fk . In fact, 27

we explicitly have rk;fk = 0 and rk;fk ?1 = vfk ?2 =vfk ?1  1. If rk;n+1  (1?)?1 holds for n some with k2  n  fk ? 2, then (B.8) and the de nition of k2 implies rk;n  (1 ? )?1  + 1 = (1 ? )?1 . Thus if fk  k2 + 1, rk;n for k2  n  fk are bounded by a constant independent of n and k. k2 is independent of n and k. Therefore rk;n for 0  n  k2 are bounded by a constant independent of n and k. If fk < k2 + 1, similar argument shows, with rk;fk = 0, that rk;n for 0  n  fk are bounded by a nite number independent of n and k. This with (B.6) and (B.7) implies the statement. 2

Theorem B.2. Let fwng, fwn0 g, n 2 Z+, be sequences in [0; 1] satisfying maxfwn ; wn0 g  Cw ?n ; n 2 Z+ ; for positive constants (independent of n) Cw and  > 1. Also let fan ; n 2 Z+ g be a sequence satisfying inf n an  2 and supn an < 1. For each k 2 Z+ consider a sequence in [0; 1]2

f(xk;n ; yk;n ) ; n = k; k ? 1;


; 0g  [0; 1]2 ; and assume that it satis es a recursive inequality an+1 ; y an+1 + w xk;n  xk;n n+1 minf1 ? xk;n +1 k;n+1 g ; +1 0 yk;n  xk;n+1 + wn+1 yk;n+1 ; n = k ? 1; k ? 2;


; 0 :

If

(B.9)

lim inf

k!1; xk;k 6=0

(

(? log xk;k )

Yk )1=k =0

a

>1

holds, then there exist positive constants C1 and C2 (independent of k) such that

maxfxk;0 ; yk;0 g  C1 exp(?C2 k2 ) ; k 2 Z+ : Proof. De ne fx~k;n ; n = k; k ? 1;


; 0g by x~k;k = xk;k and

an+1 + w ; n = k ? 1; k ? 2;


; 0 ; x~k;n = (1 ? wn+1 ) x~k;n n+1 +1 Then the recursion relation for xk;n and the assumption yk;n  1 imply xk;n  x~k;n ; 0  n  k; k  0 : fx~k;n g satis es all the assumptions of Lemma B.1 with D = 2, hence there exist positive constants C3 and k1 (independent of n and k) such that
? (B.10) xk;n  C3 sup w + exp ?2fk ?n?1 ; 0  n  fk ; k  k1 ; n

28

where (B.11)

Yk

p

fk = supfn  k ? 1 j 2 (? log xk;k )

=n+2

a > 1g + 1 :

Since (B.9) implies klim f = 1, there exists a constant k2  k1 such that for !1 k 0  n  fk =2 and k  k2 ,

?








exp ?2fk ?n?1  exp ?2fk =2?1  Cw ?fk =2  Cw ?n : This with (B.10) implies xk;n  (C3 + 1) Cw ?n , 0  n  fk =2, k  k2 . Applying this estimate to the original recursion relations and using wn  Cw ?n , wn0  Cw ?n , and an  2, we have

xk;n  Cw ?n?1 ((C3 + 1) xk;n+1 + yk;n+1 ) ; yk;n  xk;n+1 + Cw ?n?1 yk;n+1 ; 0  n  fk =2 ; k  k2 :

Iterating once, we nd maxfxk;n ; yk;n g  C5 ?n?1 maxfxk;n+2 ; yk;n+2 g ; 0  n  fk =2 ? 1 ; k  k2 ; where C5 is a positive constant independent of n and k. Iterating this [fk =4] times, where [x] is the largest integer not exceeding x, and using xk;n  1, yk;n  1, we nd (B.12) maxfxk;0 ; yk;0 g  exp

(  f

k (log C ) ? 5

4

 f 2 k

4

)

(log ) ; k  k2 :

The assumption (B.9) implies that there exist positive constants k3  k2 and

Y 0 > 1 (independent of k) such that (? log xk;k ) a > 0k , k  k3 . The de k



=0



0 k ? 1; k , k  k . Applying nition (B.11) then implies fk  min loglog 3 sup a this to (B.12), increasing constants for terms with k < k3 if necessary, we have the statement. 2

C Products of matrices with positive elements. We present an elementary theorem on the existence of a limit of normalized products of matrices with positive elements. We assume no relation among matrices in the product, such as commutativity or stationarity. We also allow the in mums of some components to be zero. 29

Theorem C.1. Let d and q be positive integers, E def = f1; 2;


; dg, and fAN , N = 1; 2; 3;


g be a sequence of d-dimensional matrices whose elements are positive and bounded, satisfying N;i;j inf (AN AN +1


AN +q?1 )ij > 0. Then for (A1


AN )ij exists, positive, and is independent = Nlim i 2 E and j 2 E , i def !1 (A1


AN )1j

of j . Proof. For N > n  0 de ne

= An+1 An+2


AN BnN def and put

B0Nij ; i 2 E ; j 2 E : =B

Nij def 0N 1j

The elementwise positivity of AN +1 and BnN imply for each i that fmin

g k Nik is increasing and fmax

g is decreasing in N , in particular, the sequence k Nik f Nij ; N = 1; 2;


g is compact. Therefore, for each i and j , and for any subsequence of positive integers there exists a further subsequence faN g such that the limit (C.1) = Nlim

> 0:

ij(a) def !1 aN ;ij exists and is positive. For 0 < n < N and i 2 E , j 2 E , put = B0nB1i BnNij : pnNij def 0N 1j

The de nition and the elementwise positivity of BnNij imply, for 0 < n < N , i; j 2 E , (C.2)

0 < pnNij < 1 ;

X

k2E

pnNkj = 1 ; Nij =

X

k2E

nik pnNkj :

We prove a couple of Lemma for pnNkj . Lemma C.2. Fix faN g, and let ij(a) be as above. If for every i; j 2 E either inf inf p > 0 or n> inf0 N>n inf+q pnNji > 0 hold, then for every i 2 E , ij(a) is n>0 N>n+q nNij independent of j . Proof. Put n = aM and N = aM 0 in (C.2). We see from (C.1) and (C.2) that for each  > 0 there is an integer M0 such that for any integers M , M 0 satisfying M 0 > M > M0 we have (C.3)

X ( ij(a) ? ik(a))paM ;a 0 ;kj <  : M k2E 30

Now suppose that the Lemma is wrong; ik(a1) < ik(a2) and ik(a1)  ij(a)  ik(a2), j 2 E . If we put j = k2 in (C.3) and keep k = k1 term in the summation we have  > ( ik(a2) ? ik(a1))paM ;aM 0 ;k2 ;k1 , while if we put j = k1 and keep k = k2 term we have  > ( ik(a2) ? ik(a1))paM ;aM 0 ;k1 ;k2 . Since  > 0 is arbitrary, these inequalities contradicts the assumption of the Lemma. 2

Lemma C.3. inf inf p n>0 N>n+q nNij

> 0 ; i 2 E; j 2 E :

Proof. Note that each pnNij is positive by (C.2). Therefore it is sucient to consider the cases where N and n are suciently large. For suciently large N ,

X

A2;i;k2 B2;N;k2;j A2;i;k2 ; A k 11 i 2 X  XA11i min p1Nij  X A A2;k1 ;k2 B2;N;k2;j A11k1 max A11k1 k2 max k1 2;k1 ;k2 k k1

1

k2

k1

where P a c we used aan inequality among non-negative numbers ai, bi,pci , i 2 E ; i . Hence inf p P bici  min inf0 N>n inf+q pnNij > 0, 1Nij > 0. If we prove n> i N> 1 b i 1Nij i i then the Lemma is proved. For suciently large N and n with N ? q > n, A B B B min A A B pnNij  min k 11k 1nki nNij  min k 11k 2kk1 n?q;n;k2 i n;n+q;ik3 : p1Nij max A B B fki g A11i A2ik1 max Bn?q;n;k2 k Bn;n+q;kk3 k k 11i 1nik nNkj Taking the in mum of both sides with respect to N and n, we see, with the assumptions of Theorem C.1, n> inf0 N>n inf+q ppnNij > 0. 2 1Nij

Let us continue the proof of the Theorem. Lemma C.2 and Lemma C.3 imply that ij(a) of (C.1) is independent of j . Fix i 2 E , and consider two subsequences of positive integers. There are subsubsequences, faN g and fbN g, for each of the subsequences respectively, such that the limits (C.4)

= Nlim

> 0; = Nlim

> 0 ; and i(b) def

i(a) def !1 bN ;ij !1 aN ;ij

exist, positive, and are independent of j . Put n = bM and N = aM 0 in (C.2); (C.5)

aM 0 ;ij =

X

k2E

bM ;ik pbM ;aM 0 ;kj ; aM 0 > bM ; i 2 E ; j 2 E : 31

The equations (C.4), (C.5), and (C.2) imply that for any positive  there exists an integer N0 such that if aM 0 > bM > N0 hold, then

! (a) (b) (a) (b) X i ? i = i ? i pbM ;aM 0 ;kj <  ; j 2 E :

k2E ( b ) ( a ) Hence i = i , which implies that the limit is independent of subsequences.

2

Positivity of the limit also follows from (C.4).

D Basic estimates on generating functions. We give an explicit formula for the generating function ga;b;i(w; h) = F~i (a; b; w; ~ (w) + w h) ; introduced in Section 3. For each i 2 E , ga;b;i (w; h) = Numi=Deni , where 3 X





O ; I W ;i =1  0 O0  Numi = ? det I 0 Wi : i def ? 1 Put Z (i) = (w)i + w S (w)ii hi , i 2 E . Then O1 = (Z (Ar); 0; 0; 0; 0; Z (Bq)), O2 = (0; Z (Ar); Z (Bq); 0; 0; 0), and O3 = (0; 0; 0; Z (Bp); Z (Bp); 0). For X 2 fA; B; D; E g and t 2 fp; q; rg, we write Xt to specify an element in E , with an 0 = O3 , X 2 fA; B; E g, O0 = O1 , and obvious rule. With this convention, OXp Dq Oi0 = O2 , otherwise. I1;Dq = 0, otherwise I1;Xt = t (Z (Xr); 0; 0; 0; 0; Z (Xp)). I2;At = t (0; Z (Ar); Z (Ap); 0; 0; 0), otherwise I2;Xt = 0. I3;Xt = 0, if X 2 fA; E g, 0 = t I3;Xq , while for t 6= q, otherwise I3;Xt = t (0; 0; 0; Z (Xq); Z (Xq); 0). IXq 0 = I1;Xt . W is a 6 dimensional matrix given by IXt W 0 =W I?(1) W (1) 0 1 0 0 Z (Bq)  BB W (1) W(1) Z (Bq) 0 0 0 C C BB 0 Z (Ap) W(2) W 0 (2) 0 0 C C: BB 0 0 W (2) W0 (2) Z (Br) 0 C C @ 0 0 0 Z (Br) W0 (2) W (2) A Z (Ap) 0 0 0 W 0 (2) W (2)
(j ) For j = 1; 2, W (j ) + 0 (j ) = W0 (j ) + (j ) = 1 ? (j )
(j ) n(j) , W (j ) = n(j )?1  (j ) n(j)=2 (j )  0 (j ) n(j)=2 (j ) , W 0 (j ) =
(j ) , where (j ) =
(j ) 0 (j ) (j ) Deni = det W +

det

n(j )?1

0

n(j )?1

32

p

n(j )+1 ? x? (j )n(j )+1 (j ) 0 (j ) , 2(j ) = (j ) + 0 (j ), and
(j )n(j) = x+ (j ) x (j ) ? , x? (j ) +  p x (j ) = 1 + (j )  (j ) (2 + (j )), (j ) = 1 ? 22( j)(j?) 2 (j ) . Finally, n(j ), (j ), 0 (j ), (j ), and 0 (j ) are given by n(1) = a ? 1, n(2) = b ? 1, (1) = 0 (1) = Z (Ap) Z (Dq), (2) = Z (Br) Z (Er), 0 (2) = Z (Bq) Z (Ep), (1) = 0 (1) = Z (Ar) + Z (Ap) Z (Dq), (2) = Z (Bp) + Z (Br) Z (Ep), 0 (2) = Z (Bq) + Z (Bq) Z (Er).

With these explicit formula, we obtain the following order estimate. De ne

Ci , i 2 E , by CAr = 1=2, CBr = CEr = 1, and Ci = 0, otherwise. Proposition D.1. For all i 2 E , w?3 Deni and w?4 (Numi ? Ci Deni) are rational in w and h, analytic at w = h = 0. We also nd by REDUCE calculation that O(w3 ) terms in Deni do not

vanish; (D.1)

lim w?3 Deni 6= 0 ; i 2 E :

w!0

The matrix A~(a; b; w) de ned in Section 3 is rational in w, and has no poles in w  0. The explicit form of A~(a; b; w = 0) given below is obtained by explicit calculation of the rst derivatives of F~ given above, using REDUCE. De ne, for notational simplicity, a matrix M (a; b) by M (a; b)ij = (b +2)2 (a + ? 1) 1 A~(a; b; w = 0)ij , i; j 2 E , and put B2 = b + 2. Then

2 2B 0 (b3 + 9b2 + 14b + 12)=12 66 (aB2 +2b)B2 (a + 1)B22 (b2 + 4b + 6)(b ? 1)=6 66 2B2 0 (b3 + 9b2 + 20b + 24)=6 6 0 b(b + 4)(b ? 1)=6 M (a; b) = 66 2(aB 0? 1)B 2 2 aB b(b2 + 4b + 7) 2 2 2 66 0 b(b + 1)B2 =4 64 2B0 0 (b3 + 9b2 + 14b + 12)=6 2

2(aB2 ? 1)B2

2aB22

(b2 + 4b + 6)(b ? 1)=3

b(b + 4)(b ? 1)=6 b(b + 5)(b + 1)=12 0 b(b2 + 6b + 11)=3 b(2b2 + 9b + 13)=12 (a ? 1)B22 b(b + 4)(b ? 1)=3 b(b + 5)(b + 1)=6 0 (b3 + 9b2 + 14b + 12)=3 b(b + 5)(b + 1)=6 0 2b(b2 + 3b + 5) (2b2 + 5b + 8)(b + 1)=2 2(a ? 1)B22 b(b + 1)B2 =2 b(b + 1)B2 =4 B22 b(b + 4)(b ? 1)=3 b(b + 5)(b + 1)=6 0 2b(b2 + 6b + 11)=3 b(2b2 + 9b + 13)=6 2(a ? 1)B22

33

3

(b2 + 10b + 12)(b ? 1)=12 b(b + 7)(b ? 1)=12 (b2 + 4b + 6)(b ? 1)=6 (2b2 + 11b + 24)(b ? 1)=12 77 77 (b2 + 10b + 12)(b ? 1)=6 b(b + 7)(b ? 1)=6 77 : b(b + 4)(b ? 1)=6 b(b + 7)(b ? 1)=6 2 2 (b + 2b + 2)(b ? 1) (2b + 3b + 8)(b ? 1)=2 77 77 b(b ? 1)B2 =4 b(b ? 1)B2 =4 5 b(b + 8)(b + 1)=6 b(b + 7)(b ? 1)=6 2 2 b(b + 6b + 11)=3 b(2b + 15b + 37)=6 All the formula in this appendix except (D.1) and the explicit form of A~ are derived by hand. At present the only proof of (D.1) is by REDUCE, but it is a `soft' result. It is straightforward to see that Proposition D.1 and (D.1) imply the estimates in Proposition 3.1 for second and third derivatives of g. The explicit form of A~ is also derived by REDUCE. The required estimates in Section 3 are (3.4), (3.5), and (3.6), among which (3.4) and (3.5) re ects the network structure of the (pre-) fractal, and (3.4) is an expectation related to one-dimensional simple random walk. It therefore suces with relatively soft estimates of A~. With these considerations, presumably, we may be able to avoid computer aided proof after all. For our purpose, rigorous derivation by REDUCE is sucient. Technical methods of derivation for the formula and estimates derived by hand in this appendix may be interesting in it's own respect, but the derivations are lengthy, and taking into account of the main interest of this paper, we omit further explanations.

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