Ising spin glass models versus Ising models: an effective mapping at high temperature III. Rigorous formulation and detailed proof for general graphs
arXiv:0706.1949 · doi:10.1088/1742-5468/2007/09/P09010
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.
25 pages, 5 figures, made statements in Sec. 10 clearer
References in corpus (4)
- Loop series for discrete statistical models on graphs
- How to Compute Loop Corrections to Bethe Approximation
- Ising spin glass models versus Ising models: an effective mapping at high temperature I. General result
- Ising spin glass models versus Ising models: an effective mapping at high temperature II. Applications to graphs and networks
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