Generalized range of slow random walks on trees

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 $\mathbb{V}$.

58 pages

References in corpus (6)

• Convergence in law of the minimum of a branching random walk
• The Seneta--Heyde scaling for the branching random walk
• A Local Limit Theorem for random walks conditioned to stay positive
• Almost sure convergence for stochastically biased random walks on trees
• Transient Random Walks in Random Environment on a Galton-Watson Tree
• The maximal potential energy of biased random walks on trees

Cited by in corpus (1)

• Coalescence in small generations for the diffusive randomly biased walk on Galton-Watson trees