Ising spin glass models versus Ising models: an effective mapping at high temperature III. Rigorous formulation and detailed proof for general graphs

TL;DR

This paper rigorously proves a mapping between Ising spin glass models and non-random Ising models on infinite-dimensional graphs, broadening understanding of critical behavior in complex graph structures.

Contribution

It introduces a new definition of graph dimensionality and provides a detailed proof of the mapping for a wide class of graphs, including those where Bethe approximation fails.

Findings

Mapping holds for all graphs satisfying the new dimensionality condition

Provides a general Nishimori law as a consequence of the mapping

Includes graphs where traditional approximations may be invalid

Abstract

Recently, it has been shown that, when the dimension of a graph turns out to be infinite dimensional in a broad sense, the upper critical surface and the corresponding critical behavior of an arbitrary Ising spin glass model defined over such a graph, can be exactly mapped on the critical surface and behavior of a non random Ising model. A graph can be infinite dimensional in a strict sense, like the fully connected graph, or in a broad sense, as happens on a Bethe lattice and in many random graphs. In this paper, we firstly introduce our definition of dimensionality which is compared to the standard definition and readily applied to test the infinite dimensionality of a large class of graphs which, remarkably enough, includes even graphs where the tree-like approximation (or, in other words, the Bethe-Peierls approach), in general, may be wrong. Then, we derive a detailed proof of the mapping for all the graphs satisfying this condition. As a byproduct, the mapping provides immediately a very general Nishimori law.