The random geometry of equilibrium phases

TL;DR

This survey explores how percolation theory applies to equilibrium statistical mechanics, covering models, phase transitions, and random interactions, highlighting the geometric aspects of equilibrium phases.

Contribution

It provides a comprehensive overview of the applications of percolation theory in understanding equilibrium phases and phase transitions in statistical mechanics.

Findings

Percolation is key to understanding phase transitions.

Random-cluster models unify percolation and spin systems.

Non-percolation implies uniqueness and exponential mixing.

Abstract

This is a (long) survey about applications of percolation theory in equilibrium statistical mechanics. The chapters are as follows: 1. Introduction 2. Equilibrium phases 3. Some models 4. Coupling and stochastic domination 5. Percolation 6. Random-cluster representations 7. Uniqueness and exponential mixing from non-percolation 8. Phase transition and percolation 9. Random interactions 10. Continuum models