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Propagation of chaos and residual dependence in Gibbs measures on finite sets

Jonas Jalowy, Zakhar Kabluchko, Matthias Löwe

Abstract

We compare a mean-field Gibbs distribution on a finite state space on $N$ spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni 1999, where marginal distributions of size $k=o(N)$ are compared to product measures.

Propagation of chaos and residual dependence in Gibbs measures on finite sets

Abstract

We compare a mean-field Gibbs distribution on a finite state space on spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni 1999, where marginal distributions of size are compared to product measures.

Paper Structure

This paper contains 4 sections, 9 theorems, 103 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['theorem:main_gen']}
  3. Classical propagation of chaos
  4. Example and extensions: the Curie-Weiss Model

Key Result

Lemma 1.1

As $N\to \infty$ it holds where the weight is given by $| w|=\sum_{j=1}^pw_j$ for $w_j=( \det(H_j)\prod_{k=1}^q M_{j,k})^{-1/2}$.

Theorems & Definitions (20)

  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • proof
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['theorem:main_gen']}
  • Remark 2.2
  • ...and 10 more