Scaling exponents for random walks on Sierpinski carpets and number of distinct sites visited: a new algorithm for infinite fractal lattices

Abstract Let $$X_1, X_2, \ldots $$ X 1 , X 2 , … be i.i.d. copies of some real random variable X. For any deterministic $$\varepsilon _2, \varepsilon _3, \ldots $$ ε 2 , ε 3 , … in $$\{0,1\}$$ { 0 , 1 } , a basic algorithm introduced by H.A. Simon yields a reinforced sequence $$\hat{X}_1, \hat{X}_2 , \ldots $$ X ^ 1 , X ^ 2 , … as follows. If $$\varepsilon _n=0$$ ε n = 0 , then $$ \hat{X}_n$$ X ^ n is a uniform random sample from $$\hat{X}_1, \ldots , \hat{X}_{n-1}$$ X ^ 1 , … , X ^ n - 1 ; otherwise $$ \hat{X}_n$$ X ^ n is a new independent copy of X. The purpose of this work is to compare the scaling exponent of the usual random walk $$S(n)=X_1+\cdots + X_n$$ S ( n ) = X 1 + ⋯ + X n with that of its step reinforced version $$\hat{S}(n)=\hat{X}_1+\cdots + \hat{X}_n$$ S ^ ( n ) = X ^ 1 + ⋯ + X ^ n . Depending on the tail of X and on asymptotic behavior of the sequence $$(\varepsilon _n)$$ ( ε n ) , we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.

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AN EFFICIENT IMPLEMENTATION OF THE EXACT ENUMERATION METHOD FOR RANDOM WALKS ON SIERPINSKI CARPETS

Fractals ◽

10.1142/s0218348x00000172 ◽

2000 ◽

Vol 08 (02) ◽

pp. 155-161 ◽

Cited By ~ 9

Author(s):

ASTRID FRANZ ◽

CHRISTIAN SCHULZKY ◽

STEFFEN SEEGER ◽

KARL HEINZ HOFFMANN

Keyword(s):

Probability Distribution ◽

Random Walks ◽

Mean Square Displacement ◽

Computation Time ◽

Scaling Exponent ◽

Exact Enumeration ◽

Time Step ◽

The Mean ◽

Dynamic Data Structure ◽

Sierpinski Carpets

In the following, we present a highly efficient algorithm to iterate the master equation for random walks on effectively infinite Sierpinski carpets, i.e. without surface effects. The resulting probability distribution can, for instance, be used to get an estimate for the random walk dimension, which is determined by the scaling exponent of the mean square displacement versus time. The advantage of this algorithm is a dynamic data structure for storing the fractal. It covers only a little bit more than the points of the fractal with positive probability and is enlarged when needed. Thus the size of the considered part of the Sierpinski carpet need not be fixed at the beginning of the algorithm. It is restricted only by the amount of available computer RAM. Furthermore, all the information which is needed in every step to update the probability distribution is stored in tables. The lookup of this information is much faster compared to a repeated calculation. Hence, every time step is speeded up and the total computation time for a given number of time steps is decreased.

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