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Interacting Particle Systems on Random Graphs

F. Capannoli, F. den Hollander

Abstract

The present overview of interacting particle systems on random graphs collects the notes of a mini-course given by the authors at the Brazilian School of Probability, 5--9 August 2024, in Salvador, Bahia, Brazil. The content is a personal snapshot of an interesting area of research at the interface between probability theory, combinatorics, statistical physics and network science that is developing rapidly.

Interacting Particle Systems on Random Graphs

Abstract

The present overview of interacting particle systems on random graphs collects the notes of a mini-course given by the authors at the Brazilian School of Probability, 5--9 August 2024, in Salvador, Bahia, Brazil. The content is a personal snapshot of an interesting area of research at the interface between probability theory, combinatorics, statistical physics and network science that is developing rapidly.

Paper Structure

This paper contains 47 sections, 36 theorems, 177 equations, 13 figures.

Table of Contents

  1. An overview: structure and goals
  2. LECTURE 1: Interacting Particle Systems on $\mathbb{Z}^d$, $d \geq 1$. General properties and three examples
  3. Spin-flip systems
  4. Shift-invariant attractive systems
  5. Three examples
  6. Partial ordering and coupling
  7. Convergence to equilibrium
  8. Properties of the three examples
  9. Example 1: Stochastic Ising Model.
  10. Example 2: Voter Model.
  11. Example 3: Contact Process.
  12. The Cox-Greven finite systems scheme
  13. $\bullet$ SIM ON THE TORUS.
  14. $\bullet$ VM ON THE TORUS.
  15. $\bullet$ CP ON THE TORUS.
  16. ...and 32 more sections

Key Result

Lemma 1.1

Let $(P_t)_{t\geq0}$ denote the semigroup of transition kernels that is associated with $\xi=(\xi_t)_{t \geq 0}$. Write $\delta_\eta P_t$ to denote the law of $\xi_t$ conditional on $\xi_0=\eta$ (which is a probability distribution on $\Omega$). Then

Figures (13)

  • Figure 1: A finite connected non-oriented graph.
  • Figure 2: Caricature picture of the free energy landscape [free energy = energy $-$ entropy].
  • Figure 3: The free energy per vertex $f_{\beta,h}(m)$ at magnetisation $m$ (caricature picture with $\mathbf{m}=m^*_-$, $\mathbf{c} = m^*$, $\mathbf{s} = m^*_+$).
  • Figure 4: Metastable regime for the parameters $\beta,h$.
  • Figure 5: Erdős-Rényi random graph (ERRG): take the complete graph with $N$ vertices and retain edges with probability $p \in (0,1)$.
  • ...and 8 more figures

Theorems & Definitions (39)

  • Lemma 1.1
  • Corollary 1.2
  • Lemma 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 2.1
  • ...and 29 more