Mark Holmes

The mean square displacement of random walk on the Manhattan lattice

Nicholas R Beaton, Mark Holmes

Journal article  |  2023  |  STATISTICS & PROBABILITY LETTERS

We give an explicit formula for the mean square displacement of the random walk on the d-dimensional Manhattan lattice after n ste..

Historical Lattice Trees

M Cabezas, A Fribergh, M Holmes, E Perkins

Journal article  |  2023  |  Communications in Mathematical Physics

We prove that the rescaled historical processes associated to critical spread-out lattice trees in dimensions d> 8 converge to his..

Coexistence of lazy frogs on ℤ

M Holmes, D Kious

Journal article  |  2022  |  Journal of Applied Probability

We study the so-called frog model on with two types of lazy frogs, with parameters respectively, and a finite expected number of d..

The Continuous-Time Lace Expansion

David Brydges, Tyler Helmuth, Mark Holmes

Journal article  |  2021  |  COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS

We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion appli..

A SHAPE THEOREM FOR THE ORTHANT MODEL

Mark Holmes, Thomas S Salisbury

Journal article  |  2021  |  ANNALS OF PROBABILITY

We study a particular model of a random medium, called the orthant model, in general dimensions d≥ 2. Each site x∈ Z d independent..

Alignment Percolation

Nicholas R Beaton, Geoffrey R Grimmett, Mark Holmes

Journal article  |  2021  |  MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in d≥ 2 dimensions..

ABSENCE OF WARM PERCOLATION IN THE VERY STRONG REINFORCEMENT REGIME

Christian Hirsch, Mark Holmes, Victor Kleptsyn

Journal article  |  2021  |  ANNALS OF APPLIED PROBABILITY

We study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex f..

Phase transitions for degenerate random environments

Mark Holmes, Thomas S Salisbury

Journal article  |  2021  |  ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS

Open-end nonparametric sequential change-point detection based on the retrospective CUSUM statistic

Mark Holmes, Ivan Kojadinovic

Journal article  |  2021  |  ELECTRONIC JOURNAL OF STATISTICS

The aim of online monitoring is to issue an alarm as soon as there is significant evidence in the collected observations to sugges..

Non-Coupling from the Past

GR Grimmett, M Holmes

Book Chapter  |  2021

The method of ‘coupling from the past’ permits exact sampling from the invariant distribution of a Markov chain on a finite state ..