Massimo Ostilli J. Stat. Mech. (2007) P09010 doi:10.1088/1742-5468/2007/09/P09010
Massimo Ostilli
Show affiliationsRecently, 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 by-product, the mapping provides immediately a very general Nishimori law.
75.10.Nr Spin-glass and other random models
75.10.Hk Classical spin models
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Issue 09 (September 2007)
Received 14 June 2007, accepted for publication 4 August 2007
Published 13 September 2007
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