# MASDOC - Mathematics and Statistics Doctoral Training Centre

### Mathematical Interests

I have finished my Ph.D supervised by Dr David Croydon as part of the MASDOC programme at the University of Warwick. I graduated from the University of Warwick in 2013 with a first class honours degree in MMORSE. My main academic interests revolve around probability theory; in particular, random walks in random environments. Most of my work to date has been on biased random walks on Galton-Watson trees conditioned to survive however I am also researching more general randomly trapped random walks. I am also interested in stochastic particle systems having done my masters dissertation in coupling results for interacting particle systems with Dr Jon Warren.

This page focusses on work done during my Ph.D, I have another webpage which I keep up to date with my current work.

### Research

I have submitted various articles in the field of random walk in random environment. More information can be found here. Below is a list of papers submitted for publication:

• A quenched central limit theorem for biased random walks on supercritical Galton-Watson trees (arXiv)
• Central limit theorems for biased randomly trapped random walks on $\mathbb{Z}$ (arXiv)
• Escape regimes of biased random walks on Galton-Watson trees. Probab. Theory Relat. Fields (2017)

In my MSc thesis I investigated the asymptotic speed of a biased random walk on a subcritical Galton-Watson tree conditioned to survive as part of a summer research project 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.

I have done a research group project on stochastic growth models with John Sylvester and Qiaochu Chen supervised by Dr Nikos Zygouras and Dr Partha Dey. In this project we showed a range of regimes for limiting shapes for a long range first passage percolation model in $\mathbb{Z}^2$.

Title: Stochastic growth models

Abstract: We consider a long-range first-passage percolation model on the two dimensional lattice $\mathbb{Z}^2$ 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 $l^1$ 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.

### Invited Talks

Random walks on Galton-Watson trees, February 2018, Markov processes on metric measure spaces and Gaussian fields workshop, University of Duisburg-Essen.

Regeneration structures for random walk models, November 2017, Statistical mechanics seminar, University of Warwick.

Limit Processes for Random Walks in Random Environments, June 2017, MASDOC Retreat, Brecon Beacons, Powys.

Random Walks on Galton-Watson Trees, September 2016, Workshop on Random Processes in Discrete Structures, University of Warwick.

Transition from annealed to quenched asymptotics for random walk in random environment, May 2016, MASDOC Retreat, Wilderhope Manor, Shropshire.

Random Walks on Galton-Watson Trees, April 2016, UK Easter Probability Meeting, Lancaster University.

Biased Random Walks on Subcritical Galton-Watson Trees, October 2015, Probability Seminar, RIMS, Kyoto University.

Stable Limit Laws in RWRE, April 2015, MASDOC-CCA Joint Workshop, University of Warwick.

Random Walks on Galton-Watson Trees Conditioned to Survive, February 2015, Postgraduate Seminar, University of Warwick.

### Events Attended

15-16 February 2018, Markov processes on metric measure spaces and Gaussian fields workshop, University of Duisburg-Essen.

25 May 2017, Recent Progress on the Geometry of Random Walks, University of Cambridge.

30 August - 02 September 2016, Workshop on Random Processes in Discrete Structures, University of Warwick.

19-24 June 2016, School and Workshop on Random Interacting Systems, University of Bath.

04-08 April 2016, UK Easter Probability Meeting, Lancaster University.

29 March - 02 April 2016, Probabilistic models - from discrete to continuous, University of Warwick.

26-29 October 2015, Stochastic Analysis on Large Scale Interacting Systems, RIMS, Kyoto University.

18-22 May 2015, Random walks on graphs and potential theory, University of Warwick.

11-15 May 2015, MASDOC Summer School: Topics in Renormalisation Group Theory and Regularity Structures, University of Warwick.

15-17 April 2015, MASDOC-CCA Joint Workshop, University of Warwick.

23-27 March 2015, YEP XII: Random walk in random environment, EURANDOM Eindhoven.

16-20 March 2015, Random Graphs, Random Trees and Applications, Isaac Newton Institute for Mathematical Sciences, Cambridge.

### Teaching Assistant Roles

2017/28 - ST202: Stochastic Processes.

2017/18 - MA258: Mathematical Analysis III.

2016/17 - ST333: Applied Stochastic Processes.

2015/16 - ST115: Introduction to Probability.

2014/15 - ST115: Introduction to Probability.

### Background

I am originally from a small village on the edge of Cambridgeshire called Fowlmere however I currently reside in Coventry. I am a keen sportsman having participated in tennis, rugby union, football and dodgeball competitions. In the latter of which, I have competed internationally for the Scotland Highlanders and won many tournaments with the Warwick Warriors. I enjoy reading classic novels; in particular nineteenth century French and Russian literature. I have also been a representative for both the postgraduate mathematics SGSLC and secretary of the MASDOC SSLC.

I have been an avid member of the Warwick University Warriors dodgeball team since 2009, have played in the first team since 2011, and held the roles of club coach, men's captain and vice captain. Many tournaments are recorded and videos uploaded to the Warwick Dodge youtube channel. Some notable matches are the final of the Yorkshire open in 2011 against Jammy Dodgers where I made my first double catch and won my first gold medal and the match against Reepham Raiders in meet 4 of DPL4 where I made double catches in consecutive games to salvage an important win. Since 2013 I have been playing internationally with the Scotland Highlanders first team. We have entered nine tournaments in this time winning the 2016 SIx Nations, coming third in 2014 Six Nations and second in 2013 Six Nations, 2015 Six Nations, 2013 Euros, 2014 Euros, 2015 Euros and 2016 Euros. We also came fifth in the first ever world cup in Manchester.

### Contact

I can be contacted via email through a dot bowditch at warwick dot ac dot uk or found in the office D0.05 of the Zeeman Building most days.