Phase transitions in stochastic systems

Historical Lattice Trees

M Cabezas, A Fribergh, M Holmes, E Perkins

2023-01-01

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

2022-09-28

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

2021-09-04

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

2021-05-01

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

2021-03-01

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

2021-02-01

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

Non-Coupling from the Past

GR Grimmett, M Holmes

2021-01-01

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

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

Mark Holmes, Ivan Kojadinovic

2021-01-01

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

Phase transitions for degenerate random environments

Mark Holmes, Thomas S Salisbury

2021-01-01

CURIOSITIES REGARDING WAITING TIMES IN POLYA'S URN MODEL

Norbert Henze, Mark Holmes

2020-08-01

Consider an urn, initially containing b black and w white balls. Select a ball at random and observe its colour. If it is black, s..

Biased Random Walk on the Trace of Biased Random Walk on the Trace of ...

David Croydon, Mark Holmes

2020-04-01

We study the behaviour of a sequence of biased random walks (X(i))i≥0 on a sequence of random graphs, where the initial graph is Z..

On the range of lattice models in high dimensions

Mark Holmes, Edwin Perkins

2020-04-01

We investigate the scaling limit of the range (the set of visited vertices) for a general class of critical lattice models, starti..

A paradox for expected hitting times

M Holmes, PG Taylor

2020-03-06

We prove a counterintuitive result concerning the expected hitting/absorption time for a class of Markov chains. The “paradox” alr..

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Heather Mitcheltree, Mitchell Ransome

2019-12-01

A Monotonicity Property for Once Reinforced Biased Random Walk on Z(d)

Mark Holmes, Daniel Kious

2019-01-01

We study once-reinforced biased random walk on Zd. We prove that for sufficiently large bias, the speed v(β) is monotone decreasin..

On the speed of once-reinforced biased random walk on trees

Andrea Collevecchio, Mark Holmes, Daniel Kious

2018-01-01

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unrei..

Proof of the WARM whisker conjecture for neuronal connections

Mark Holmes, Victor Kleptsyn

2017-04-01

This paper is devoted to the study of the so-called WARM reinforcement models that are generalisations of Pólya's urn. We show tha..

Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

Mark Holmes, Thomas S Salisbury

2017-01-01

We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on ℤd. Standard conditions for ballis..