Random walks in random environment on trees and multiplicative chaos

TL;DR

This paper analyzes the behavior of random walks in random environments on regular trees, linking their classification to multiplicative chaos, and localizing the critical point for ergodicity and transience.

Contribution

It establishes a connection between random walks in random environments on regular trees and multiplicative chaos, providing new insights into their classification and critical points.

Findings

Classification of ergodicity and transience based on geometric properties of the environment matrix

Localization of the critical point for the random walk behavior

Conjecture linking chaos solutions to null recurrence of the walk

Abstract

We study random walks in a random environment on a regular, rooted, coloured tree. The asymptotic behaviour of the walks is classified for ergodicity/transience in terms of the geometric properties of the matrix describing the random environment. A related problem, with only one type of vertices and quite stringent conditions on the transition probabilities but on general trees has been considered previously in the literature LyoPem. In the presentation we give here, we restrict the study of the process on a regular graph instead of the irregular graph used in LyoPem. The close connection between various problems on random walks in random environment and the so called multiplicative chaos martingale is underlined by showing that the classification of the random walk problem can be drawn by the corresponding classification for the multiplicative chaos, at least for those situations where both problems have been solved by independent methods. The chaos counterpart of the problem we considered here has not yet been solved. The results we obtain for the random walk problem localise the position of the critical point. We conjecture that the additional conditions needed for the chaos problem to have non trivial solutions will be the same as the ones needed for the random walk to be null recurrent.