Christophe Sabot

On a family of self-interacting processes interpolating between the non-directed edge reinforced random walk and the directed edge reinforced random walk.

In this talk I will present a family of self-interacting processes interpolating between the (directed) Edge Reinforced Random Walk (ERRW) and the non-directed Edge Reinforced Random Walk. By the standard representation of Polya urns, the latter one corresponds also to a random walk in random environment with independent transition probabilities at each site, with Dirichlet distributions. These two processes have rather different behaviours on Z^d (e.g. in d>2, phase transition between recurrence and transience for the first one and transience for the second one) and are investigated by rather different methods. The new process we consider, called the *-ERRW, is non reversible but has some type of symmetry with respect to an involution * of the graph. A generalisation of the Vertex Reinforced Jump Process, the *-VRJP, can be associated with the *-ERRW. Contrary to the standard VRJP, the *-VRJP is not exchangeable after time change, which leads to several new phenomena. However, it becomes exchangeable after a randomization of the initial local times. I will present a few results on the representation of these processes, in particular a generalisation of the representation of the VRJP in terms of random Schrödinger operators. This leads to new identities between integrals, which are rather mysterious even in simplest cases.

Based on joint works with Sergio Bacallado and Pierre Tarrès.