Abstract

Random Walks in random Dirichlet Environment (RWDE) is the model of random walks in random environment with i.i.d. Dirichlet distributed transition probabilities at each site. With this specific choice of environment the annealead law is the directed Edge Reinforced Random Walk. Besides, under some condition on the weights the time-reversed walk is again a RWDE . This was used e.g. in dimension d>2, to prove the existence of an invariant measure for the process viewed from the particle, absolutely continuous with respect to the static law. We will present a new identity for RWDE , inspired by the Vertex Reinforced Jump Process (VRJP), more precisely by its non-reversible generalization, the *-VRJP. From this identity we will deduce some sufficient conditions in 2D for the existence or non-existence of an absolutely continuous invariant measure for the process viewed from the particle. Joint with Adrien Perrel.