Some results on history dependent stochastic processes

Edge reinforced random walk (ERRW) and vertex reinforced jump processes are history dependent stochastic process, where the particle tends to come back more often on sites it has already visited in the past. For a particular scheme of reinforcement these processes are mixtures of reversible Markov chains whose mixing measure can be related to a non-linear sigma model introduced in the context of random matrices. I will give an overview on these models and explain some recent results in joint work with F. Merkl and S. Rolles.