Biased random walk on critical Galton-Watson trees conditioned to survive

TL;DR

This paper studies biased random walks on critical Galton-Watson trees conditioned to survive, showing they share universality with one-dimensional trapping models with slowly-varying tails, including functional limit theorems and extremal aging.

Contribution

It establishes a connection between biased random walks on critical Galton-Watson trees and one-dimensional trapping models, demonstrating shared universality class and related limit theorems.

Findings

Shared universality class with trapping models

Functional limit theorems involving extremal processes

Demonstration of extremal aging phenomena

Abstract

We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.