Random walk with memory

There are a large number of different modifications and variants of the usual symmetrical random walk ~RW!. 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 Diaconis introduced reinforced random walk ~RRW!. This walk, opposite to SAW, prefers earliest visited paths. Pemantle 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. Davis 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 wk be an increasing sequence of non-negative numbers. Let (Xn) be a random motion on Z. If some interval was traversed k-times, then its weights is wk . If Xn5i , then the probability that Xn115i21 or Xn115i11 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 ` wk ,` . This result was generalized to RRW sequence type on the d-dimensional lattice by Sellke. 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 APZ. Let p1 ,. . . ,pN be the probabilities of choosing one of the adjacent points A1 ,. . . ,AN . Assume that we choose the point Ai0. 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 p18 , . . . ,pN8 . The values of p18 , . . . ,pN8 depend on the previous values p1 ,. . . ,pN and i0 . 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