Finite Markov chains in statistical mechanics and combinatorics. Finite Markov chains can be viewed as random walks in a finite set. In applications, this set often consists of certain combinatorial objects whose typical properties are to be understood. If the set is large, obtaining exact solutions to such problems is generally infeasible. Markov chains can provide a highly efficient method to generate randomised approximations in such cases, but only if they equilibrate at a rate that grows sl ....Finite Markov chains in statistical mechanics and combinatorics. Finite Markov chains can be viewed as random walks in a finite set. In applications, this set often consists of certain combinatorial objects whose typical properties are to be understood. If the set is large, obtaining exact solutions to such problems is generally infeasible. Markov chains can provide a highly efficient method to generate randomised approximations in such cases, but only if they equilibrate at a rate that grows slowly with the size of the set of objects under study. The project will study several classes of Markov chains that have been developed to study a number of notoriously difficult problems in statistical mechanics and combinatorics, and determine under what conditions they provide efficient approximation schemes.Read moreRead less
ARC Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights. In today's world, massive amounts of data in a variety of forms are collected daily from a multitude of sources. Many of the resulting data sets have the potential to make vital contributions to society, business and government, as well as impact on international developments, but are so large or complex that they are difficult to process and analyse using traditional tools. The aim of this ....ARC Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights. In today's world, massive amounts of data in a variety of forms are collected daily from a multitude of sources. Many of the resulting data sets have the potential to make vital contributions to society, business and government, as well as impact on international developments, but are so large or complex that they are difficult to process and analyse using traditional tools. The aim of this Centre is to create innovative mathematical and statistical models that can uncover the knowledge concealed within the size and complexity of these big data sets, with a focus on using the models to deliver insight into problems vital to the Centre's Collaborative Domains: Healthy People, Sustainable Environments and Prosperous Societies.Read moreRead less
Phase transitions in stochastic systems. 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 c ....Phase transitions in stochastic systems. 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.Read moreRead less