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A Course in Interacting Particle Systems

Jan M. Swart

TL;DR

A Course in Interacting Particle Systems develops a comprehensive, rigorous scaffold for studying countable lattices of locally interacting Markov processes. It lays out a generator-based construction with graphical (Poisson-point) representations, analyzes mean-field limits on complete graphs, and connects finite- and infinite- lattice dynamics through Feller- and Lyapunov-theoretic methods. The book surveys canonical models (voter, contact, Ising–Potts, exclusion, branching/coalescing) and extends to ergodicity and scaling limits, highlighting universality, phase transitions, and the rich interplay between probabilistic and analytical techniques. Together, these tools enable both qualitative understanding and quantitative control of complex spatial stochastic systems, with broad applications in physics, biology, and social dynamics.

Abstract

These lecture notes give an introduction to the theory of interacting particle systems. The main subjects are the construction using generators and graphical representations, the mean field limit, stochastic order, duality, and the relation to oriented percolation. An attempt is made to give a large number of examples beyond the classical voter, contact and Ising processes and to illustrate these based on numerical simulations.

A Course in Interacting Particle Systems

TL;DR

A Course in Interacting Particle Systems develops a comprehensive, rigorous scaffold for studying countable lattices of locally interacting Markov processes. It lays out a generator-based construction with graphical (Poisson-point) representations, analyzes mean-field limits on complete graphs, and connects finite- and infinite- lattice dynamics through Feller- and Lyapunov-theoretic methods. The book surveys canonical models (voter, contact, Ising–Potts, exclusion, branching/coalescing) and extends to ergodicity and scaling limits, highlighting universality, phase transitions, and the rich interplay between probabilistic and analytical techniques. Together, these tools enable both qualitative understanding and quantitative control of complex spatial stochastic systems, with broad applications in physics, biology, and social dynamics.

Abstract

These lecture notes give an introduction to the theory of interacting particle systems. The main subjects are the construction using generators and graphical representations, the mean field limit, stochastic order, duality, and the relation to oriented percolation. An attempt is made to give a large number of examples beyond the classical voter, contact and Ising processes and to illustrate these based on numerical simulations.

Paper Structure

This paper contains 60 sections, 106 theorems, 680 equations, 36 figures.

Table of Contents

  1. Introduction
  2. General set-up
  3. The voter model
  4. The contact process
  5. Ising and Potts models
  6. Phase transitions
  7. Variations on the voter model
  8. The exclusion process
  9. Branching and coalescing particles
  10. Periodic behavior
  11. Continuous-time Markov chains
  12. Finite state space
  13. The embedded Markov chain
  14. Generator construction
  15. Lyapunov functions
  16. ...and 45 more sections

Key Result

Proposition 2.3

Let $(P_t)_{t\geq 0}$ be the semigroup of a Markov process with finite state space $S$ and generator $G$. Then for each probability measure $\mu$ on $S$ and functions $f,g\colon S\to{\mathbb R}$, one has

Figures (36)

  • Figure 1: Four snapshots of a two-dimensional voter model with periodic boundary conditions. Initially, the types of sites are i.i.d. Time evolved in these pictures is 0, 1, 32, and 500.
  • Figure 2: Four snapshots of a three-dimensional voter model with periodic boundary conditions. Initially, the types of sites are i.i.d. Time evolved in these pictures is 0, 4, 32, and 250.
  • Figure 3: Four snapshots of a two-dimensional contact process. Initially, only a single site is infected. The infection rate is 2, the death rate is 1, and time evolved in these pictures is 1, 5, 10, and 20.
  • Figure 4: Survival probability of the one-dimensional contact process.
  • Figure 5: Four snapshots of a $q=4$, $\beta=1.2$ Potts model with Glauber dynamics and periodic boundary conditions. Initially, the types of sites are i.i.d. Time evolved in these pictures is 0, 4, 32, 500.
  • ...and 31 more figures

Theorems & Definitions (106)

  • Proposition 2.3: Covariance formula
  • Theorem 2.4: Convergence to equilibrium
  • Theorem 2.7: Generator construction
  • Lemma 2.8: First jump decomposition
  • Lemma 2.9: The backward equation
  • Lemma 2.10: Comparison principle
  • Proposition 2.11: Uniqueness of solutions
  • Theorem 2.12: Sufficient conditions for nonexplosiveness
  • Lemma 2.13: Exponential bound
  • Lemma 2.14: Bounded jump rates
  • ...and 96 more