Phase transitions in stochastic systems

Grant number: FT160100166 | Funding period: 2017 - 2021

Completed

Abstract

This project aims to understand models of physical and biological phenomena in the presence of uncertainty/randomness. Such models often exhibit phase transitions if a variable defining the model is modified. For example, a population explosion can occur if the average number of offspring per individual is larger than one, while macroscopic defects can occur in a material if the density of microscopic defects is larger than some threshold. This research could lead to strategies for directing physical and biological systems towards preferred states or phases, and better prediction of adverse events such as fracturing of Antarctic sea ice.

Related publications (11)

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A SHAPE THEOREM FOR THE ORTHANT MODEL

Mark Holmes, Thomas S Salisbury

2021-05-01

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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..

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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 ..

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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..

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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..

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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..

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A paradox for expected hitting times

M Holmes, PG Taylor

2020-03-01

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

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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..

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

A Collevecchio, M Holmes, D Kious

2018-01-01

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

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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..

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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..

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