We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton-Watson trees with a countable number of types, thus extending a result of G. Miermont in [18] on multitype Galton-Watson trees with finitely many types.MSC2010 subject classification 60J80, 60F17

Summary. Erdős and Révész [15] initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Gaton-Watson tree. We prove that there is some parameter κ ∈ (1, ∞] such that the set of the favorite sites of the biased walk is almost surely bounded in the case κ ∈ (2, ∞], tight in the case κ = 2, and oscillates between a neighborhood of the root and the boundary of the range in the case κ ∈ (1, 2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case κ ∈ (2, ∞]. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees.

No abstract