Random walk with memory - Semantic Scholar

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 40, NUMBER 6

JUNE 1999

Random walk with memory Ryszard Rudnicki Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8, 40-013 Katowice, Poland and Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

Marek Wolfa) Institute of Theoretical Physics, University of Wrocław, PI. Maxa Borna 9, PL-50-204 Wrocław, Poland

~Received 16 June 1998, accepted for publication 17 November 1998! A reinforced random walk on the d-dimensional lattice is considered. It is shown that this walk is equivalent to an iterated function system ~IFS!. Criteria for the existence of limit cycles are given. Numerical results and conjectures about the quantitative behavior of the walk are stated. © 1999 American Institute of Physics. @S0022-2488~99!00905-6#

I. INTRODUCTION

There are a large number of different modifications and variants of the usual symmetrical random walk ~RW!.1–3 Let us mention only Levy flights, biased diffusions, self-avoiding walk ~SAW for short!, etc. Let us confine ourselves to the random walks on the discrete lattices. In SAW a walking particle is choosing its trajectory in such a way that it does not step down onto the already visited site. If a particle runs into such a node that all neighboring sites were already visited, it stops. In Ref. 4 the interacting RW was discussed in which the parameter 0, p,` has influenced probabilities of visiting a given site and p51 corresponds to usual RW. For p→` this RW goes on into the SAW. In 1987 Coppersmith and Diaconis5 introduced reinforced random walk ~RRW!. This walk, opposite to SAW, prefers earliest visited paths. Pemantle6 discussed a related process on trees and proved that this process is equivalent to a random walk in a random environment. He also gave criteria for transience and recurrence of RRW. Davis7 considered a variety of types of RRW on the integers Z. One of them was RRW of sequence type. This process is defined in the following way. Let w k be an increasing sequence of non-negative numbers. Let (X n ) be a random motion on Z. If some interval was traversed k-times, then its weights is w k . If X n 5i, then the probability that X n11 5i21 or X n11 5i11 is proportional to the weights at time n of the intervals (i21,i) and (i,i11). Davis proved that the moving point visits a finite number of integers and eventually oscillates between two adjacent integers if and only if ( `k50 w 21 k ,`. This result was generalized to RRW sequence type on the d-dimensional lattice by Sellke.8 In this paper we consider another type of reinforced random walk on the d-dimensional lattice. The random point moves according to the following reinforcement convention. Let the moving point be found at time t5n at a certain point APZn . Let p 1 ,..., p N be the probabilities of choosing one of the adjacent points A 1 ,...,A N . Assume that we choose the point A i 0 . If after some time the moving point returns to A, then the probabilities that at the next step it can be found at the adjacent points are equal to p 18 ,...,p N8 . The values of p 18 ,..., p N8 depend on the previous values p 1 ,..., p N and i 0 . We assume that the probability of choosing a given path will increase when it was already traversed and probabilities of remaining paths emanating from a given site will decrease. In other words, the fact that some sites were already visited will be remembered. The memory of passing particular edges will be encoded in the change of probabilities. At some time a!

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© 1999 American Institute of Physics

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J. Math. Phys., Vol. 40, No. 6, June 1999

R. Rudnicki and M. Wolf

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the probability of going in some direction from a given site will reach almost 1, while probabilities to go in other directions will be practically zero. It will result in closed paths: a random walker will oscillate between a few sites with practically zero probability to escape from such a limiting cycle. We will treat such a final behavior as stopping of the random walk. Our walk is not Markovian because the probability of choosing any direction changes in time. If we extend the phase space by adding the distributions of probabilities of passing particular edges, we obtain a Markov process which is also an iterated function system.9 In Refs. 10 and 11 were introduced self-attracting diffusions: processes attracted by their own trajectories. It is interesting that these processes and our walk have similar features. For example, self-attracting diffusions are not Markovian, but jointly with their occupation measures are Markov processes. Moreover, their trajectories converge almost surely. The paper is organized as follows. First we define our random walk and the notion of the limit cycle is introduced. Next we prove the theorem that this walk reaches the limit cycle. In Sec. IV the result of the Monte Carlo simulations for a particular ‘‘memorizing’’ function are presented. These computer experiments allow us to make some conjecture about the quantitative behavior of some characteristics of the walk.

II. MATHEMATICAL MODEL

A. Description of the random walk with memory

Let Z denote the set of all points of the d-dimensional Euclidean space which have integer coordinates, i.e., Z is the d-dimensional lattice. A point moves randomly over this lattice. It starts at point 05~0,...,0!. If at time t5n the moving point can be found at a certain point x 5(x 1 ,...,x d ), then at the time t5n11 it can be found at one of the N52 d adjacent points y 5(y 1 ,...,y d ), where y i 5x i 11 or y i 5x i 21 for i51,2,...,d. By K we denote the set of all possible ‘‘steps’’ during the walk, i.e., K5 $ ~ x,y ! PZ3Z: u y i 2x i u 51

for i51,...,d % .

If z5(x,y)PK, then the points x and y are, respectively, the beginning and the end of the step z. Let S5 $ 1,21 % d be the set of all N steps to the nearest neighbors. If xPZ, sPS, and y5x1s, then (x,y)PK. At time t50 the probability of choosing of any adjacent point equals 2 2d . During the walk the point ‘‘memorizes’’ its path in the following way. Assume that the moving random walker can be found at some time t5n at a certain point xPZ. Let p x,x1s , sPS, be the probabilities that at the time t5n11 it can be found at one of the adjacent points x1s, sPS. Assume that we choose the point x1s 0 to shift the particle from the point x. If after some time the moving point returns to x, then the probabilities that at the next step it can be found at the adjacent points are equal to 8 8 p x,x1s , sPS. The numbers p x,x1s are related to the previous values p x,x1s in the following way. 8 ) sPS, and Let px 5(p x,x1s ) sPS , px8 5(p x,x1s

H

P5 px P @ 0, 1 # S:

(

sPS

J

p x,x1s 51 .

Then p8x 5 f s 0 ~ px ! ,

~1!

where f s 0 :P→P is a continuous function. If zPK and z5(x,y), we will often write p z instead of p x,y .


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B. Iterated function system

The state of the random walk with memory is described at any time t by the position of the moving point and the probability p z for every zPK. Let P denote the set of all admissible distributions of probabilities p z , zPK, i.e.,

H

P5 pP @ 0, 1 # K:

(

sPS

p x,x1s 51

J

for each xPZ .

Then the phase space is the set X5Z3P. Since px 5(p x,x1s ) sPS for every xPZ, we have P 5PZ and X5Z3PZ. Now, we define an iterated function9 system on the phase space X. It consists of N transformations T s :X→X, sPS. These transformations are defined as follows. Let xPZ and sPS be given. Then T s (x, p)5(x1s, p 8 ), where pz8 5

H

pz ,

if z 1 Þx,

f s ~ pz ! ,

if z 1 5x,

for each z5(z 1 ,z 2 )PK. If (x, p) is the state of the random walk at a time t and if the next position of the moving point is x1s, then T s (x,p) is the next state of the random walk. The probability that at a point x 5(x, p) we choose the transformation T s is equal p s ( x )5p x,x1s . C. Markov process on X

Now we construct a Markov process corresponding to the iterated function system given in Sec. II B. The phase space X is a metric space with some metric r defined as follows. The set K is countable, that is, K5 $ z 1 ,z 2 ,z 3 ,... % , where $ z n % nPN , z n PK, is the sequence of all possible steps. If xPX, yPX, then x5(u,p z 1 , p z 2 ,...) and y5(u 8 ,p z8 ,p z8 ,...), where u, u 8 PZ and p z k ,p z8 1 2 k P @ 0, 1 # for each kPN. The metric r is given by `

r ~ x,y ! 5 u u2u 8 u 1

(

k51

2 2k u p z k 2 p z8 u . k

Let B be the s-algebra of Borel subsets of X. For any xPX and APB we set I ~ x,A ! 5 $ sPS:T s ~ x ! PA % ,

P ~ x,A ! 5

(

sPI ~ x,A !

p s~ x ! .

Then P(x,A) is a transition probability function, i.e., ~a! ~b!

for each xPX the function A° P(x,A) is a probabilistic measure and for each APB the function x° P(x,A) is B-measurable.

Since the space X is s-compact, there exists a homogeneous Markov process $ j n % `n50 which corresponds to the transition function P(x,A). 12 It means that we have some probability space ~V, A, Prob! and a sequence $ j n % `n50 of random elements j n :V→X such that the sequence j n is a Markov process and Prob ~ j n11 PA u j n 5x ! 5 P ~ x,A ! for each xPX, APB, n>0. Since the initial state of the system is x 0 5(0,22d ,22d ,...), we assume that j 0 5x 0 . We assume that the probability space ~V, A, Prob! is complete, i.e., if A is a measurable set and Prob (A)50, then every subset of A is measurable.




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D. Limit cycle

A sequence (u 0 ,u 1 ,...,u m21 ) of different elements of Z is called a cycle if there exists a sequence (s 0 ,s 1 ,...,s m21 ) of elements of S such that u k11 5u k 1s k for k50,...,m21, where u m 5u 0 . Let P 0 :X→Z be the operator given by P 0 (u,p)5u. We say that a sequence (x n ) `n50 of elements of X has a limit cycle if there exist a cycle (u 0 ,u 1 ,...,u m21 ) and an integer n 0 >0 such that for every n>n 0 we have P 0 (x n )5u k , where k5n(mod m). Let c 0 (t)5l and c n (t)5 c + c n21 (t). Now we can formulate the following theorem. Theorem 1: Let c:@0, 1#→@0, 1# be a continuous nondecreasing function such that c~1!51 and `

( n50

F S DG 12 c n

1 N

,`.

~2!

We assume that there exists a continuous function c:@0, 1#→@0, 1# such that f s,s ~ px ! > c ~ p x,x1s !

~3!

for sPS, where f s,s is the s-th coordinate of f s . Then there exists a measurable subset V 0 ,V such that Prob (V 0 )51 and for each v PV 0 the sequence $ j n ( v ) % `n50 has a limit cycle. The proof of Theorem 1 is given in Sec. III. Remark 1: Let P z , zPK, be the operator P z :X→ @ 0, 1 # given by P z (u,p)5 p z . Assume that the sequence $ j n ( v ) % `n50 has the limit cycle (u 0 ,u 1 ,...u m21 ). Then from Theorem 1 it follows that limn→` P z „j n ( v )…51 for each z5(u k ,u k11 ), k50,...,m21. Remark 2: If c:@0, 1#→@0, 1# is a continuous nondecreasing function such that c (x).x for xP(0,1) and c8~1!.1, then c satisfies ~2! III. PROOF OF THEOREM 1 A. Boundedness of trajectories

The thread of the proof of Theorem 1 is as follows. First we check that almost all paths are bounded. From this it follows that a point performing a random walk returns infinitely often to some points of the lattice Z. Then we show that if a point uPZ is visited infinitely often, then after some time the random walker chooses a fixed adjacent point to u. This implies that the random walk has a limit cycle. Let h n ( v )5P 0 „j n ( v )…. Then the random variable h n describes the position of the moving point at time t5n. Proposition 1: For almost all v the sequence $ h n ( v ) % is bounded. We precede the proof of Proposition 1 with the following lemmas. Lemma 1: Let w (t)5P `n50 c n (t). Then w (t).0 for t>1/N and lim w ~ t ! 51.

~4!

t→1

Proof: Since c is a nondecreasing function, from ~2! it follows that

w~ t !>w

SD

1 ,` N

for tP

F G

1 ,1 . N

~5!

Let «.0 be given. Since w (1/N).0 there is an integer n 0 such that `

) n5n

`

c ~ t !> n

0

) n5n

cn 0

SD

1 .12« N

for tP

F G

1 ,1 . N

~6!


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The function c is continuous and c~1!51. This implies that there is d.0 such that n 0 21

)

n50

c n ~ t ! .12«

for tP @ 12 d ,1# .

~7!

From ~6! and ~7! it follows that w (t).(12«) 2 for tP @ 12 d , 1 # . Consequently, limt→1 w (t)51.h Lemma 2: Let xPZ and yPZ be two adjacent points. Denote by A the event that the point x is visited for the first at time t5n 0 and the point y was not visited earlier. Let B5 $ v PV: h n 0 12i21 ~ v ! 5y,

h n 0 12i ~ v ! 5x,

for i51,2,...% .

Then the conditional probability Prob (B u A) satisfies

F S DG

Prob ~ B u A ! > w

1 N

2

.

Proof: Let B 0 5A and B k 5 $ v PV: h n 0 12i21 ~ v ! 5y,

h n 0 12i ~ v ! 5x,

for i51,2,...,k % .

If v PB k ùA, then at each time n 0 c f k21 s,s N N k

f s 8 ,s 8

> c 2 f k22 s,s

1 N

>¯> c k

SD

1 , N

1 1 >ck . N N

Consequently,

F S DG

Prob ~ B k11 u B k ùA ! > c k From ~8! it follows that

1 N

2

F) S DG k

Prob ~ B k11 ùA ! >Prob ~ A !

i50

ci

~8!

.

1 N

2

.

~9!

If k→`, then we obtain

F S DG

Prob ~ BùA ! >Prob ~ A ! w

1 N

2

,

and finally




F S DG

Prob ~ B u A ! > w

1 N

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2

.0.

h

Proof of Proposition 1: For xPRd we set i x i 5max$ u x i u :i51,...,d % .

For k51,2,..., we define C k 5 $ v PV:supi h n v !i >k % . n

According to Lemma 2,

F S DG

Prob ~ C k11 u C k ! @ w (1/N) # 2 the moving point visits only x and y at any time t.n 0 . This implies ~10!. From ~10! it follows that

H F S DG J

Prob ~ C k ! < 12 w

1 N

2 k21

.

~11!

Let C5ù `k51 C k . Since a trajectory h n ( v ) is unbounded if and only if v PC, from ~11! it follows that almost all trajectories are bounded. h B. Stabilization of directions

From Proposition 1 it follows that almost all trajectories are bounded. Consequently, the moving point visits some points of the lattice Z infinitely often. Let a point xPZ be given. By A we denote the event that the point x is visited infinitely often. For any v PA we denote by $ k n ( v ) % `n51 successive times of visits at point x. Let x n ( v ) be the adjacent point to x visited at time t5k n ( v )11. We show that for almost every v PA there exists a point y( v )PZ such that x n ( v )5y( v ) for n.n 0 ( v ). The process of choosing the adjacent points can be described as an iterated function system ( f s ) sPS on the space P and the probability that at the point px PP we choose the transformation f s 0 equals p x,s 0 . Indeed, let us assume that we visit the point x and let px 5(p x,s ) sPS be the distribution of probability of choosing adjacent points x1s, sPS. If we choose the point x1s 0 , then at the next visit at x, p8x 5 f s 0 (px ) is the new distribution of probability of choosing adjacent points. Since x is a given point we will write p instead of px and p s instead of p s,x . Let $ z n % `n51 be a homogeneous Markov process on the phase space P corresponding to the iterated function system ( f s ) sPS . The transition probability function for the process ( z n ) is given by the formula P ~ p,A ! 5

( ps ,

sPI

where I5 $ sPS: f s ~ p! PA % .

~12!

Denote by x1s n ( v ) the adjacent point chosen at time t5n. Then z 1 5(1/N,...,1/N), z n11 5 f s n ( z n ), and Prob ~ s n 5s u z n 5p! 5 p s .

~13!

for pPP, n51,2,..., and sPS.


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Proposition 2: For almost every v there is s( v )PS and an integer n 0 ( v ) such that s n ( v ) 5s( v ) for n.n 0 ( v ). We precede the proof of Proposition 2 by a lemma. Let Pse 5 $ pPP:p s >« % , Pe 5 ø Pse . sPS

Lemma 3: Let P n (p,A) be the n-step transition probability function. Then for every d,1 and pPP we have lim P n ~ p,Pd ! 51.

~14!

n→`

Proof: Let «,1 be given. First we check that for pPP« we have P ~ p,Pc ~ « ! ! >«.

~15!

Indeed, if pPP« , then for some s we have pPPs« and ps >«. Consequently, f s (p)PPcs («) and inequality ~15! follows immediately from ~12!. From ~15! it follows that P i11 ~ p,Pc i11 ~ « ! ! >

E

Pc i ~ « !

P ~ q,Pc i11 ~ « ! ! P i ~ p,dq! > c i ~ « ! P i ~ p,Pc i ~ « ! ! ,

which gives n21

P n ~ p,Pc n ~ « ! ! >

)

c k~ « ! > w ~ « !

k50

~16!

for pPP« .

If «>1/N, then limn→` c n («)51. From ~16! it follows that for every d,1, «>1/N, and pPP« we have lim inf P n ~ p,Pd ! > w ~ « ! .

~17!

n→`

If pPP, then pPP1/N and, consequently, lim infP n ~ p,Pd ! > w n→`

SD

1 . N

~18!

Since P n1m ~ p,Pd ! 5

E

P

P n ~ q,Pd ! P m ~ p,dq! 5

E

Pd

P n ~ q,Pd ! P m ~ p,dq! 1

the inequalities ~17! and ~18! imply lim infP n1m ~ p,Pd ! > w ~ d ! P m ~ p,Pd ! 1 w n→`

SD

X

E

P\Pd

P n ~ q,Pd ! P m ~ p,dq! ,

S DC

1 m 1 P ~ p,P\Pd ! 5 w ~ d ! 2 w N N

P m ~ p,Pd ! 1 w

SD

1 . N ~19!

Set

a ~ d ! 5lim infP n ~ p,Pd ! . n→`



Then from ~19! it follows


X

a~ d !> w~ d !2w

S DC 1 N

a~ d !1w

SD

1 . N

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~20!

Since a~d! is a nonincreasing function there exists the limit limd →1 a ( d )5 a 0 . According to Lemma 1, limd →1 w ( d )51. A passage to the limit d→1 in inequality ~20! gives

X S DC S D

a 0 > 12 w

1 N

1 . N

a 01 w

From ~21! we conclude that a 0 >1 and ~14! holds. Proof of Proposition 2: Let d P(1/N,1) be a given number. Since

~21! h

Prob ~ z n PPd ! 5 P n21 ~ z 1 ,Pd ! , from Lemma 3 it follows that there exists n 0 such that Prob ~ z n 0 PPd ! > d .

~22!

Let A5 $ v : z n 0 PPd % and A s 5 $ v : z n 0 PPsd % . Then A5ø sPSA s and the sets A s , sPS, are pair disjoint. From ~13! it follows that m

Prob ~ s n 0 11 5s,...,s n 0 1m 5s u A s ! >

)

i50

c i~ d ! > w ~ d ! .

~23!

Let B s 5 $ v :s n ~ v ! 5s

for n>n 0 ~ v ! % ,

B5 ø B s . sPS

Inequality ~23! implies that Prob ~ B s u A s ! > w ~ d ! and consequently Prob ~ B s ! > w ~ d ! Prob ~ A s ! .

~24!

The sets A s , sPS, are pair disjoint and the sets B s , sPS, are pair disjoint. From ~22! and ~24! we obtain Prob ~ B ! > w ~ d ! Prob ~ A ! > w ~ d !~ 12 d ! . Letting d→0 we have Prob (B)51, which completes the proof.

h

C. Existence of the limit cycles

Now we are ready to complete the proof of Theorem 1. According to Proposition 1 almost every trajectory is bounded and goes through some point u 0 PZ infinitely often. Let D be a bounded subset of Z and u 0 PZ be a given point. Denote by A 0 the subset of V which consists of all vPV such that the trajectory $ h n ( v ) % is contained in D and goes through u 0 infinitely often. According to Proposition 2, for almost every v PA 0 there exists s( v )PS such that the random walker going through u 0 chooses the direction s( v ) for sufficiently


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TABLE I. The numbers RW of the cycle for a few values of a.

2 4 6 8 10 12 14 16 18

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

9554 397 38 7 0 0 1 0 0

9555 410 29 5 0 0 1 0 0

9592 363 38 3 1 1 0 0 0

9640 329 23 5 0 0 0 0 0

9645 316 27 6 3 1 0 0 0

9633 343 19 1 2 0 0 0 0

9723 247 17 4 0 0 0 0 1

8973 199 10 6 2 0 0 2 0

large times. Now, we can divide the set A 0 into N disjoint subsets B 1 ,...,B N in such a way that in each set B k the step (u 0 ,u 1 ) is determined uniquely. Denote one of these sets by A 1 . Then we can divide the set A 1 into N subsets related to the next step (u 1 ,u 2 ), etc. After some steps the moving point returns to u 0 and in this way we obtain a limit cycle. h

IV. NUMERICAL SIMULATIONS A. Details of the studied models

The theorem proved in previous sections gives only general information about the RW with memory. To gain some insights into the more quantitative characteristics of RW with memory we have performed Monte Carlo simulations for the function f s ~ x ! 5 c a ~ x ! 5x @ 22 a 2 ~ 12 a ! x # ,

~25!

where a is a parameter from the interval ~0,1!. Since for a51 the function c a (x) is equal to the identical mapping, we expect that for a→1, the RW with memory will tend to the usual symmetrical RW. In particular, for a'1 it should never fall into the limiting cycle. In other words some critical slowing down in reaching the limit cycle will occur for a'1.

FIG. 1. The plot of the histograms of the number of steps performed by random walkers before the stop of the process for a sample of the values of a.




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FIG. 2. The plot of the dependence of the parameter r in ~26! obtained from the moments of actual data. Remarkably, this figure suggests that r does not depend on a.

B. Obtained results

We have performed the simulations only on the two-dimensional, square lattice of the size 130031300. In each node of the lattice we have stored probabilities of making a step in one of the four directions. Initially all p z were set to be equal 41. The random walker started from the origin of the lattice and after each step the probabilities were updated according to Eq. ~1!. A given particular simulation of the RW was finished when one p z reached the value of 0.999 or else the total number of steps was equal to 2 000 000. We have imposed the periodic boundary conditions on the RW and we recorded the facts of crossing by RW the edges of the torus. There were rare cases of such events, most of them occurred, of course, for larger values of a. We have performed simulations for a in the range ~0.8, 0.94!. In the subrange ~0.8, 0.91! a was changed with the step Da50.01, while in the subinterval ~0.91, 0.94! with the step Da50.001, because the number of steps performed by random walker before the stop was increasing very rapidly with growing a. We did not continue to larger values of a, because the number of steps needed to stop the RW was too large. For each a there were 10 000 separate random walks performed. We have stored the number of steps N at which for the first time one of the probabilities reached p z 50.999. The path of RW falls into the cycle ~see Sec. II D! and the length of the limiting trajectory was also stored. Table I gives a sample of this data for a few values of a for the length of the cycle 2, 4,..., 18—larger cycles have occurred very randomly. The numbers in this table do not sum up to 10 000 because some RW had limit cycles larger than 18, and for large a rare samples did not fall into the limit cycle in less than 2 000 000 steps. These lengths of cycles do not follow the Poisson distribution and we do not have any conjecture describing these numbers. The numbers of steps, for each a, varied considerably from one sample RW to another. For example, for a50.8 there was a RW which stabilized after N min563 steps, while the largest number of steps was N max54114. This gap between smallest and largest number of steps needed to stop RW increased with a, for example, for a50.93 the minimal and maximal number of steps before RW stopped was N min51034 and N max5590 176, respectively. We claim that the number of steps N is governed by the gamma distribution with parameters r, b: g r, b ~ x ! 5 b r x r21 e 2 b x /G ~ r ! ,

~26!

where G(r) is a generalization of the factorial:


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FIG. 3. The plot of the dependence of the parameter b in ~26! obtained from the moments of actual data.

G~ r !5

E

`

0

e 2t t r21 dt.

~27!

In Fig. 1 we present plots of the histograms of the number of steps for a few of values of a. The size of the bins was 1000, so the y axis gives the number of random walks with the number of steps in the range (10003k,10003k11000). In Figs. 2 and 3 the values of the fitted parameters r and b for all investigated values of a are shown. Remarkably, the parameter r takes values around 1.72 and it seems not to depend on a. It is probably linked with the special choice of the function ~25!. Despite the large fluctuation of N between different realizations of RW, there seems to be a simple formula describing the median m value of N. Here m is defined as such a value of N that the same number of sample random walks stopped in smaller than m, as well in larger than m steps. Since for a51 the RW with memory passes into the usual RW, N should diverge to infinity for

FIG. 4. The plot of dependence of the median m~a! versus u51/(12 a ) is shown. The solid line presents the least-square fit to the points obtained from the Monte Carlo simulations, represented by circles, under the assumption that fit is made by the exponential function.




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a→1, thus we guessed that m is a function of 1/~12a!. Hence, in Fig. 4 the plot of the median m versus u51/(12 a ) is shown. This figure suggests that m~a! grows exponentially with u—the dashed line presents the exponential fit to the actual values obtained by the least-square method: m ~ a ! ;exp

S D

1 . 12 a

~28!

Summarizing, the Monte Carlo simulations suggest that there seems to be strict, quantitative rules governing the behavior of some characteristics of the RW with memory for the function c a (x). ACKNOWLEDGMENTS

This research was supported by the State Committee for Scientific Research ~Poland! Grant No. 2 P03A 042 09 ~RR! and No. 2 P302 057 07 ~MW!. F. Spitzer, Principles of Random Walk ~Springer-Verlag, New York, 1976!. J. W. Haus and K. W. Kehr, Phys. Rep. 150, 263 ~1987!. 3 S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 ~1987!. 4 H. E. Stanley, K. Kang, S. Redner, and R. L. Blumberg, Phys. Rev. Lett. 51, 1223 ~1983!. 5 Two talks by P. Diaconis at the 1987 Midwest Probability Conference, unpublished. 6 R. Pemantle, ‘‘Phase transition in reinforced random walk and RWRE on trees,’’ Ann. Prob. 16, 1229 ~1988!. 7 B. Davis, Probab. Th. Rel. Fields 84, 203 ~1990!. 8 T. Sellke, ‘‘Reinforced random walk on d-dimensional integer lattice,’’ Tech. Rept. No. #94-26, Dept. of Statistics, Purdue Univ. 9 M. F. Barnsley, Fractals Everywhere ~Academic, New York, 1988!. 10 M. Cranston and Y. Le Jan, Math. Ann. 303, 87 ~1995!. 11 O. Raimond, Probab. Th. Rel. Fields 107, 177 ~1997!. 12 E. B. Dynkin, Markov Processes, Vol. I ~Springer-Verlag, New York, 1965!. 1 2
