Publications

Title: Biased random walk on supercritical percolation: Anomalous fluctuations in the ballistic regime

Year: 2021

With: David Croydon

Journal: Electron. J. Probab.

Abstract: We study biased random walk on the infinite connected component of supercritical percolation on the d dimensional integer lattice for d≥2. For this model, Fribergh and Hammond showed the existence of an exponent γ such that: for γ<1, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by γ; whereas for γ>1, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of the bias, a central limit theorem when γ>2. In this article, we explain how Fribergh and Hammond's percolation estimates further allow it to be established that for 1<γ<2 the fluctuations about the mean are of an anomalous polynomial order, with exponent given by 1/γ.

Title: The two-dimensional random field Ising model

Year: 2022

With: Rongfeng Sun

Journal: Ann. Probab.

Abstract: In this paper we construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Furthermore, we show that almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman.

Title: Non-Gaussian fluctuations of randomly trapped random walks

Year: 2021

Journal: A. Appl. Probab.

Abstract: In this paper we consider the one-dimensional, biased, randomly trapped random walk when the trapping times have infinite variance. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton-Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

Title: Differentiability of the speed of biased random walks on Galton-Watson trees

Year: 2020

With: Yuki Tokushige

Journal: ALEA Lat. Am. J. Probab. Math. Stat.

Abstract: We prove that the speed of a λ-biased random walk on a supercritical Galton-Watson tree is differentiable for λ such that the walk is ballistic and obeys a central limit theorem, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.

Title: Central limit theorems for biased randomly trapped random walks on Z

Year: 2019

Journal: Stochastic Process. Appl.

Abstract: We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have finite second moment. In a quenched environment, an environment dependent centring is determined which is necessary to achieve a central limit theorem. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive and prove a tight bound on the bias required to obtain such limiting behaviour.

Title: A quenched central limit theorem for biased random walks on supercritical Galton-Watson trees

Year: 2018

Journal: J. Appl. Probab.

Abstract: In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

Title: Escape regimes of biased random walks on Galton-Watson trees

Year: 2018

Journal: Probab. Theory Relat. Fields

Abstract: We study biased random walk on subcritical and supercritical Galton-Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballistic phase.

PhD Thesis: Supervised by Dr David Croydon

Title: Biased randomly trapped random walks and applications to random walks on Galton-Watson trees

Abstract: In this thesis we study biased randomly trapped random walks. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive. This application was initially considered model in its own right.

We prove conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. We also study the regime in which the walk is sub-ballistic; in this case we prove convergence to a stable subordinator. Furthermore, we study the fluctuations of the walk in the ballistic but sub-diffusive regime. In this setting we show that the walk can be properly centred and rescaled so that it converges to a stable process.

The biased random walk on the subcritical GW-tree conditioned to survive fits suitably into the randomly trapped random walk model; however, due to a lattice effect, we cannot obtain such strong limiting results. We prove conditions under which the walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. In these cases the trapping is weak enough that the lattice effect does not have an influence; however, in the sub-ballistic regime it is only possible to obtain converge along specific subsequences.

We also study biased random walks on infinite supercritical GW-trees with leaves. In this setting we determine critical upper and lower bounds on the bias such that the walk satisfies a quenched invariance principle.

MSc Thesis: Supervised by Dr David Croydon

Title: Biased random walks on subcritical Galton-Watson trees conditioned to survive

Abstract: In this thesis we survey known results concerning the limiting behaviour of biased random walks on supercritical and critical Galton-Watson trees conditioned to survive and extend them to the subcritical case. We start by giving a proof that there exists a well-defined probability measure over such non-extinct subcritical trees and that these exhibit a unique backbone with subcritical trees as leaves. We show that the speed exists a.s. for any bias and is positive if and only if the bias belongs to some determined region depending only on the mean and variance of the offspring distribution of the tree. We then consider this as a directed trap model to determine the speed along the backbone in terms of the bias of the walk and moments of the offspring distribution up to second order.

Research group project: With John Sylvester and Qiaochu Chen supervised by Dr Nikos Zygouras and Dr Partha Dey.

Title: Stochastic growth models

Abstract: We consider a long-range first-passage percolation model on the two dimensional lattice under a specific class of distributions supported away from 0. We show that in the critical and supercritical cases that the limiting shape is a suitably scaled ball. Moreover, we show that in the subcritical case a limiting shape exists and that under some assumptions this deterministic shape has a flat piece which coincides with that of the nearest neighbour model.