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Factorizations of relative entropy using stochastic localization

Pietro Caputo, Zongchen Chen, Daniel Parisi

TL;DR

The paper addresses entropy factorization for Gibbs spin systems by coupling stochastic localization with covariance controls to obtain strong approximate Shearer inequalities at high temperature. The main approach reduces spins to binary form, uses a stochastic localization process to drive toward a product-measure regime, and derives explicit entropy-contraction bounds that yield tight mixing-time estimates for arbitrary block dynamics, including unbounded-degree graphs and critical regimes. It provides a unifying framework that applies to general q-valued spins and extends to multicomponent systems, yielding tensorization results under weak inter-component interactions. The work delivers both concrete quantitative bounds and structural insights, bridging renormalization-group ideas with stochastic localization to obtain optimal or near-optimal results across several classical models, such as Ising and Potts, in tree-uniqueness and critical settings, with practical implications for mixing times and sampling algorithms.

Abstract

We derive entropy factorization estimates for spin systems using the stochastic localization approach proposed by Eldan and Chen-Eldan, which, in this context, is equivalent to the renormalization group approach developed independently by Bauerschmidt, Bodineau, and Dagallier. The method provides approximate Shearer-type inequalities for the corresponding Gibbs measure at sufficiently high temperature, without restrictions on the degree of the underlying graph. For Ising systems, these are shown to hold up to the critical tree-uniqueness threshold, including polynomial bounds at the critical point, with optimal $O(\sqrt n)$ constants for the Curie-Weiss model at criticality. In turn, these estimates imply tight mixing time bounds for arbitrary block dynamics or Gibbs samplers, improving over existing results. Moreover, we establish new tensorization statements for the Shearer inequality asserting that if a system consists of weakly interacting but otherwise arbitrary components, each of which satisfies an approximate Shearer inequality, then the whole system also satisfies such an estimate.

Factorizations of relative entropy using stochastic localization

TL;DR

The paper addresses entropy factorization for Gibbs spin systems by coupling stochastic localization with covariance controls to obtain strong approximate Shearer inequalities at high temperature. The main approach reduces spins to binary form, uses a stochastic localization process to drive toward a product-measure regime, and derives explicit entropy-contraction bounds that yield tight mixing-time estimates for arbitrary block dynamics, including unbounded-degree graphs and critical regimes. It provides a unifying framework that applies to general q-valued spins and extends to multicomponent systems, yielding tensorization results under weak inter-component interactions. The work delivers both concrete quantitative bounds and structural insights, bridging renormalization-group ideas with stochastic localization to obtain optimal or near-optimal results across several classical models, such as Ising and Potts, in tree-uniqueness and critical settings, with practical implications for mixing times and sampling algorithms.

Abstract

We derive entropy factorization estimates for spin systems using the stochastic localization approach proposed by Eldan and Chen-Eldan, which, in this context, is equivalent to the renormalization group approach developed independently by Bauerschmidt, Bodineau, and Dagallier. The method provides approximate Shearer-type inequalities for the corresponding Gibbs measure at sufficiently high temperature, without restrictions on the degree of the underlying graph. For Ising systems, these are shown to hold up to the critical tree-uniqueness threshold, including polynomial bounds at the critical point, with optimal constants for the Curie-Weiss model at criticality. In turn, these estimates imply tight mixing time bounds for arbitrary block dynamics or Gibbs samplers, improving over existing results. Moreover, we establish new tensorization statements for the Shearer inequality asserting that if a system consists of weakly interacting but otherwise arbitrary components, each of which satisfies an approximate Shearer inequality, then the whole system also satisfies such an estimate.

Paper Structure

This paper contains 18 sections, 18 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Spin system
  4. Entropy factorizations
  5. Block dynamics and entropy contractions
  6. Main results
  7. Generalizations to a multicomponent system
  8. Proof of Theorem \ref{['mainthm2']}, Theorem \ref{['th:SpInd']}, and Theorem \ref{['th:critical-Ising']}
  9. Reduction to binary spins
  10. Stochastic localization process
  11. Entropic stability
  12. Covariance estimate
  13. Stochastic localization and block dynamics
  14. Proof of Theorem \ref{['mainthm2']} and Corollary \ref{['cor2-Ising']}
  15. Proof of Theorem \ref{['th:SpInd']} and Theorem \ref{['th:critical-Ising']}
  16. ...and 3 more sections

Key Result

Theorem 2.2

Suppose that $\Gamma$ is positive semidefinite and that $\lambda(\Gamma)\le 1-\delta$ where $\delta \in [0,1]$. For all $\alpha\in\mathcal{W}$, for any choice of the self potentials $\psi_i:[q]\mapsto {\mathbb R}$ in spinsys, for all functions $f\ge 0$, where That is, $\mu$ satisfies the strong approximate Shearer inequality with constant $C$.

Theorems & Definitions (36)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • ...and 26 more