Self-interacting Random Walks

A self-interacting random walk is a random process evolving in an environment depending on its past behaviour.

The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.

Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system.

The purpose of our talk is to present our recent results on the subject [1,3,4,6].

Bibliography

[1] M. Bena