Abstract
Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. In parallel, similar models of random processes in a random environment have been studied. In this article we focus on a model of random walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992). Our point of view is a bit different yet, as we consider a very general way of constructing random trees with random transition probabilities on them. We prove an analogue of R. Lyons and R. Pemantle's recurrence criterion in this setting, and we study precisely the asymptotic behavior, under restrictive assumptions. Our last result is a generalization of a result of Y. Peres and O. Zeitouni (2006) concerning biased random walks on Galton-Watson trees.
Highlights
Introduction and statement of resultsModels of random walks in a random environment were introduced at first by Chernov in 1967 ([6]) in order to study biological mechanisms
We introduce the model of random walk in a random environment
We introduce a new law on trees, with particular properties
Summary
The distribution of the random variable (N (e), A(e1), A(e2), ...) is q, We will always assume m := E[N (e)] > 1, ensuring that the tree is infinite with a positive probability. We introduce the model of random walk in a random environment. Given a marked tree T , we set for x ∈ T ∗, xi a child of x, ω(x, xi) = 1+. Morever we set ω(x, y) = 0 whenever d(x, y) = 1, It is easy to check that (ω(x, y))x,y∈T is a family of non-negative random variables such that,. Twill be called “the environment”, and we call “random walk on T ” the Markov chain (Xn, T ) defined by X0 = e and. We call “annealed probability” the probability MT = MT ⊗ T taking into account the total alea
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