Scaling limit of the random walk on a Galton-Watson tree with regular varying offspring distribution

TL;DR

This paper studies the scaling limits of a random walk on a Galton-Watson tree with offspring distribution having a heavy tail, showing convergence to a stable Lévy process and associated continuum tree.

Contribution

It establishes the convergence of the height function and trace of the walk to continuous limits, linking discrete random walks to stable Lévy processes and continuum trees.

Findings

Convergence of the height function to a spectrally positive stable Lévy process.

Convergence of the walk's trace to a continuum tree.

Results hold for offspring distributions with regular varying tails of order .

Abstract

We consider a random walk on a Galton-Watson tree whose offspring distribution has a regular varying tail of order $\kappa\in (1,2)$. 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 trace of the walk towards the continuum tree coded by the latter continuous-time height process.