Central limit theorems for biased randomly trapped random walks on [formula omitted

Adam Bowditch

doi:10.1016/j.spa.2018.03.017

Abstract

We prove CLTs for biased randomly trapped random walks in one dimension. By considering a sequence of regeneration times, we will establish an annealed invariance principle under a second moment condition on the trapping times. In the quenched setting, an environment dependent centring is necessary to achieve a central limit theorem. We determine a suitable expression for this centring. As our main motivation, we apply these results to biased walks on subcritical Galton–Watson trees conditioned to survive for a range of bias values.

Highlights

We investigate biased randomly trapped random walks (RTRWs) on Z and apply the results to subcritical Galton–Watson trees conditioned to survive
The purpose of the RTRW is to generalise models such as the Bouchaud trap model and provide a framework for studying random walks on other random graphs in which trapping naturally occurs such as biased random walks on percolation clusters and random walk in random
As in the randomly trapped random walk case, we use P(·) = ∫ PTρ (·)P(dT ) for the annealed law. This is the model of the biased random walk on a subcritical GW-tree conditioned to survive which is the focus of Theorems 4–6

Summary

Introduction

We investigate biased randomly trapped random walks (RTRWs) on Z and apply the results to subcritical Galton–Watson trees conditioned to survive. As in the randomly trapped random walk case, we use P(·) = ∫ PTρ (·)P(dT ) for the annealed law This is the model of the biased random walk on a subcritical GW-tree conditioned to survive which is the focus of Theorems 4–6. A technique is developed in [6] that can be used to extend an annealed invariance principle to a quenched result This is applied in [27] to prove a quenched functional central limit theorem for the walk on the supercritical tree when the offspring distribution has exponential moments and no deaths. This will be shown in [8]

A law of large numbers and functional central limit theorem

Quenched central limit theorem

Subcritical Galton–Watson trees

The speed of the walk

An annealed functional central limit theorem

A quenched central limit theorem