Generalized range of slow random walks on trees

TL;DR

This paper investigates the generalized range of slow biased random walks on trees, analyzing how the walk's trajectory interacts with the underlying branching potential, revealing diverse behaviors and new theoretical insights.

Contribution

It introduces a general framework for studying the range of slow biased random walks on trees, including detailed examples and analysis of their interactions with branching potentials.

Findings

Diverse behaviors of the walk's range depending on the potential and bias.

New theoretical results on the volume of the trace of the walk.

Illustrative examples demonstrating different interaction regimes.

Abstract

In this work, we are interested in the set of visited vertices of a tree $\mathbb{T}$ by a randomly biased random walk $\mathbb{X}:=(X_n,n \in \mathbb{N})$. The aim is to study a generalized range, that is to say the volume of the trace of $\mathbb{X}$ with both constraints on the trajectories of $\mathbb{X}$ and on the trajectories of the underlying branching random potential $\mathbb{V}:=(V(x), x \in \mathbb{T})$. Focusing on slow regime's random walks (see [HS16b], [AC18]), we prove a general result and detail examples. These examples exhibit many different behaviors for a wide variety of ranges, showing the interactions between the trajectories of $\mathbb{X}$ and the ones of $\mathbb{V}$.