Scaling limit of the subdiffusive random walk on a Galton-Watson tree in random environment

TL;DR

This paper studies a subdiffusive random walk on a Galton-Watson tree in a random environment, showing its height function converges to a stable Lévy process, revealing the scaling limits of such stochastic processes.

Contribution

It establishes the convergence of the renormalized height and range of the walk to continuous processes, providing new insights into the scaling limits in random environments.

Findings

Height function converges to a spectrally positive stable Lévy process.

Range of the walk converges to a real tree coded by the Lévy process.

Results describe the asymptotic behavior of subdiffusive walks on random trees.

Abstract

We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable L\'evy process, jointly with the convergence of the renormalised range of the walk towards the real tree coded by the latter continuous-time height process.