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The critical temperature $T_{cr}$(Ising) is DS-computable

Senya Shlosman

TL;DR

The paper proves that Dobrushin-Shlosman (DS) uniqueness conditions precisely delineate the supercritical regime of the $d$-dimensional Ising model, establishing that the supremum of finite-box DS thresholds $eta_V$ equals the model's critical inverse temperature $eta_{ ext{cr}}(d)$. It introduces dependence coefficients via Kantorovich (KR) distance to quantify how boundary spins influence interior spins, and uses a Ding-Song-Sun covariance inequality to connect this boundary influence to decay of correlations. Consequently, the critical temperature becomes constructive: one can compute $T_{ ext{cr}}(d)$ to arbitrary precision by checking finite-box $C_V$ conditions. The results align with and extend previous constructive approaches, showing DS criteria exhaust the phase-transition boundary for the Ising model.

Abstract

We show that the Dobrushin-Shlosman conditions CV for the uniqueness of the Gibbs state provide the exact value for the critical temperature of the d-dimensional Ising model.

The critical temperature $T_{cr}$(Ising) is DS-computable

TL;DR

The paper proves that Dobrushin-Shlosman (DS) uniqueness conditions precisely delineate the supercritical regime of the -dimensional Ising model, establishing that the supremum of finite-box DS thresholds equals the model's critical inverse temperature . It introduces dependence coefficients via Kantorovich (KR) distance to quantify how boundary spins influence interior spins, and uses a Ding-Song-Sun covariance inequality to connect this boundary influence to decay of correlations. Consequently, the critical temperature becomes constructive: one can compute to arbitrary precision by checking finite-box conditions. The results align with and extend previous constructive approaches, showing DS criteria exhaust the phase-transition boundary for the Ising model.

Abstract

We show that the Dobrushin-Shlosman conditions CV for the uniqueness of the Gibbs state provide the exact value for the critical temperature of the d-dimensional Ising model.

Paper Structure

This paper contains 8 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. DS uniqueness
  3. Main result
  4. The critical temperature $T_{cr}\left( d\right)$ is constructive
  5. Constructive uniqueness
  6. KROV distance
  7. Dependence coefficients
  8. Proof of the Theorem \ref{['M']}.

Key Result

Theorem 2

$\left( DS\right)$ If for some finite $V\subset\mathbb{Z}^{d}$ the interaction $\mathcal{U}$ satisfies the $C_{V}$ condition, then it has unique Gibbs state. Moreover, this state has exponentially decaying correlations.

Theorems & Definitions (4)

  • Definition 1
  • Theorem 2
  • Conjecture 3
  • Theorem 5