Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ferromagnetic Ising Model on multiregular random graphs

Diego Alberici, Pierluigi Contucci, Emanuele Mingione, Filippo Zimmaro

Abstract

A family of multispecies Ising models on generalized regular random graphs is investigated in the thermodynamic limit. The architecture is specified by class-dependent couplings and magnetic fields. We prove that the magnetizations, neighbours correlations and free energy converge to suitable functions evaluated at the solution of a belief propagation fixed point equation. In absence of magnetic fields, a phase transition is identified and the corresponding critical parameters are determined by the spectral radius of a low-dimensional matrix.

Ferromagnetic Ising Model on multiregular random graphs

Abstract

A family of multispecies Ising models on generalized regular random graphs is investigated in the thermodynamic limit. The architecture is specified by class-dependent couplings and magnetic fields. We prove that the magnetizations, neighbours correlations and free energy converge to suitable functions evaluated at the solution of a belief propagation fixed point equation. In absence of magnetic fields, a phase transition is identified and the corresponding critical parameters are determined by the spectral radius of a low-dimensional matrix.
Paper Structure (5 sections, 17 theorems, 88 equations)

This paper contains 5 sections, 17 theorems, 88 equations.

Table of Contents

  1. Definitions and Main Results
  2. Preliminaries
  3. General results on trees
  4. From trees to graphs
  5. Comments

Key Result

Theorem 1.2

Consider a random $\mathbf{k}$-regular graph $G_N$ with $n$ classes $C_1,...,C_n$. Consider an Ising model on $G_N$ with all non-negative external fields associated to the classes $h_i$, $i= 1,...,n$, and ferromagnetic (positive) couplings $\beta_{ij}$ depending only on the classes of the interactin where $F_\beta(y) = \tanh^{-1}( \tanh(\beta) \tanh(y))$ and $\{\bar{z}_i^{(j)}\}_{i,j=1,...,n}$ is

Theorems & Definitions (43)

  • Definition 1.1
  • Remark
  • Theorem 1.2
  • Theorem 2.1: GKS inequalities
  • Remark
  • Theorem 2.2: FKG inequalities
  • Remark
  • Theorem 2.3: GHS inequalities
  • Lemma 2.4
  • Remark
  • ...and 33 more