Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2007-06-13
J. Stat. Mech. (2007) P09010
Physics
Condensed Matter
Disordered Systems and Neural Networks
25 pages, 5 figures, made statements in Sec. 10 clearer
Scientific paper
10.1088/1742-5468/2007/09/P09010
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.
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