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

Abstract

Recently, it has been shown that, when the dimension of a graph turns out to beinfinite-dimensional in a broad sense, the upper critical surface and the correspondingcritical behavior of an arbitrary Ising spin glass model defined over such a graph can beexactly mapped on the critical surface and behavior of a non-random Ising model. A graphcan be infinite-dimensional in a strict sense, like the fully connected graph, or in a broadsense, as happens on a Bethe lattice and in many random graphs. In this paper, we firstlyintroduce our definition of dimensionality which is compared to the standard definition andreadily 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 otherwords, the Bethe–Peierls approach), in general, may be wrong. Then, we derive adetailed proof of the mapping for all the graphs satisfying this condition. As aby-product, the mapping provides immediately a very general Nishimori law.