2007/06/30 by Massimo Ostilli
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Critical dimension #Curse of dimensionality #Dimension (graph theory) #Ising model #Lattice (music) #Markov Chains and Monte Carlo Methods #Random graph #Spin glass #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1088/1742-5468/2007/09/p09010
published as J. Stat. Mech. (2007) P09010 · 25 pages, 5 figures, made statements in Sec. 10 clearer
arxiv created 2007/08/01 · openalex publication_date 2007/09/14 · arxiv updated 2011/11/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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 by-product, the mapping provides immediately a very general Nishimori law.