Statistical physics studies systems consisting of many interacting components, where simple local interactions can produce rich large-scale behaviour. A central theme is the appearance of phase transitions, where a small change in a parameter might lead to a sudden qualitative change in the behaviour of the system.

Models of critical phenomena

An example of a phase transition from everyday life is that of water. At temperatures above 0° C it is liquid, while at temperatures below 0° it becomes solid ice. This drastic difference is in spite of the fact that changing the temperature does not change the underlying physics of how the water molecules interact with each other.

In order to understand the example of this phase transition and many others, physicists and mathematicians have introduced a wide variety of simplified models. The water/ice example above leads to models such as Coulomb gases. Other important models are percolation (modelling random networks) and the Ising model (modelling magnetization in metals). A common feature of these models is that one samples random configurations and, rather than keeping track of each particle, studies the behaviour of a typical particle and the large-scale properties of the system.

Geometry, scaling, and criticality

Mathematically, these models give rise to many challenges. A first question is whether the model exhibits a phase transition. If so, the next goal is to study the behaviour and geometry of the system at and near the point where the phase transition occurs (the so-called critical point). This entails, in particular, understanding the geometry of the structures that form and their scaling limits, which are often described by quantum field theories.

However, phase transitions and critical phenomena are just some of the topics of interest in statistical physics. The unifying goal is to understand how seemingly complex and chaotic interactions at a microscopic scale lead to deterministic, predictable behaviour at a macroscopic scale.

Members of ProbLaB with research in these areas include:

• Dr Antal Járai, Senior Lecturer; has studied Abelian sandpiles and related models of self-organised criticality, mainly through connections with uniform spanning trees and loop-erased random walks
• Daniel Kious, Reader; has studied random walks in random and dynamic environments, with connections to interacting particle systems and phase transitions
• Dr Cécile Mailler, Reader; has studied the zero-range model and preferential attachment models
• Dr Avi Mayorcas, Lecturer; has studied quantum field theories that describe the scaling limits of various statistical physics models
• Dr Christoforos Panagiotis, Lecturer; has studied various spin systems, such as the Ising model, using techniques from percolation theory
• Professor Tim Rogers, Professor; has studied how large-scale order emerges in complex random systems, particularly on networks and in spatial settings
• Dr Florian Schweiger, Lecturer; main research interest is the study of the interfaces between different phases in models from statistical physics