Random walks in Dirichlet environment: an overview

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights $(\alpha_i)_{i=1, \ldots, 2d}$, one for each direction of $\Bbb{Z}^d$. In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we give a state of the art on this model and several sketches of proofs presenting the core of the arguments. We also present new computation of the large deviation rate function for one dimensional RWDE.Comment: 35 page

A.S. convergence for infinite colour P\'olya urns associated with stable random walks

We answer Problem 11.1 of Janson arXiv:1803.04207 on P\'olya urns associated with stable random walk. Our proof use neither martingales nor trees, but an approximation with a differential equation.Comment: 8 page

Degrees in random mm-ary hooking networks

The theme in this paper is a composition of random graphs and P\'olya urns. The random graphs are generated through a small structure called the seed. Via P\'olya urns, we study the asymptotic degree structure in a random $m$-ary hooking network and identify strong laws. We further upgrade the result to second-order asymptotics in the form of multivariate Gaussian limit laws. We give a few concrete examples and explore some properties with a full representation of the Gaussian limit in each case. The asymptotic covariance matrix associated with the P\'olya urn is obtained by a new method that originated in this paper and is reported in [25].Comment: 21 pages, 5 figure