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Free energy equivalence between mean-field models and nonsparsely diluted mean-field models

Manaka Okuyama, Masayuki Ohzeki

TL;DR

The paper addresses whether nonsparsely diluted mean-field spin systems with Bernoulli connectivity exhibit the same thermodynamic free energy as their corresponding mean-field models. It employs an interpolation method to couple the diluted model to the standard mean-field model and derives a bound on the derivative of the interpolating pressure, showing $\frac{dA_N(t)}{dt}=\mathcal{O}(\alpha^{-1/2} N^{-b(p-1)/2})$; under $0<b\le1$ and $\alpha N^{b(p-1)}\to\infty$ this bound forces $A_N(1)-A_N(0)\to0$, proving $f_{\text{MF}}=f_{\text{dMF}}$ for both ferromagnetic and spin-glass cases. The results extend Bovier and Gayrard’s dense Curie–Weiss equivalence to arbitrary discrete spins, $p$-body interactions, and intermediate dilution regimes, for both Gaussian and bounded discrete couplings. The findings imply that many thermodynamic properties of nonsparsely diluted systems can be inferred from the corresponding mean-field models, including universal ground-state behavior in dense limits and finite-temperature equivalence under Bernoulli dilution, while noting potential breakdowns for non-Bernoulli distributions.

Abstract

We studied nonsparsely diluted mean-field models that differ from sparsely diluted mean-field models, such as the Viana--Bray model. When the existence probability of each edge follows a Bernoulli distribution, we rigorously prove that the free energy of nonsparsely diluted mean-field models with appropriate parameterization coincides exactly with that of the corresponding mean-field models in ferromagnetic and spin-glass models composed of any discrete spin $S$ in the thermodynamic limit. Our results is a broad generalization of the result of a previous study [Bovier and Gayrard, J. Stat. Phys. 72, 643 (1993)], where the densely diluted mean-field ferromagnetic Ising model (diluted Curie--Weiss model) with appropriate parameterization was analyzed rigorously, and it was proven that its free energy was exactly equivalent to that of the corresponding mean-field model (Curie--Weiss model).

Free energy equivalence between mean-field models and nonsparsely diluted mean-field models

TL;DR

The paper addresses whether nonsparsely diluted mean-field spin systems with Bernoulli connectivity exhibit the same thermodynamic free energy as their corresponding mean-field models. It employs an interpolation method to couple the diluted model to the standard mean-field model and derives a bound on the derivative of the interpolating pressure, showing ; under and this bound forces , proving for both ferromagnetic and spin-glass cases. The results extend Bovier and Gayrard’s dense Curie–Weiss equivalence to arbitrary discrete spins, -body interactions, and intermediate dilution regimes, for both Gaussian and bounded discrete couplings. The findings imply that many thermodynamic properties of nonsparsely diluted systems can be inferred from the corresponding mean-field models, including universal ground-state behavior in dense limits and finite-temperature equivalence under Bernoulli dilution, while noting potential breakdowns for non-Bernoulli distributions.

Abstract

We studied nonsparsely diluted mean-field models that differ from sparsely diluted mean-field models, such as the Viana--Bray model. When the existence probability of each edge follows a Bernoulli distribution, we rigorously prove that the free energy of nonsparsely diluted mean-field models with appropriate parameterization coincides exactly with that of the corresponding mean-field models in ferromagnetic and spin-glass models composed of any discrete spin in the thermodynamic limit. Our results is a broad generalization of the result of a previous study [Bovier and Gayrard, J. Stat. Phys. 72, 643 (1993)], where the densely diluted mean-field ferromagnetic Ising model (diluted Curie--Weiss model) with appropriate parameterization was analyzed rigorously, and it was proven that its free energy was exactly equivalent to that of the corresponding mean-field model (Curie--Weiss model).
Paper Structure (6 sections, 1 theorem, 29 equations)

This paper contains 6 sections, 1 theorem, 29 equations.

Table of Contents

  1. Introduction
  2. Models and result
  3. Proof of Theorem 1
  4. Case where $J_{d,i_1\cdots i_p}$ follows any bounded discrete probability distribution
  5. Case where $J_{d,i_1\cdots i_p}$ follows the Gaussian distribution
  6. Discussions

Key Result

Theorem 1

Let $\alpha \in (0,1]$ and $b\in (0,1]$ be functions of $N$ such that $\alpha N^{b(p-1)}\to \infty$ is $N\to\infty$. Then,

Theorems & Definitions (2)

  • Theorem 1
  • Remark 2