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A general form of certain mean field models for SPIN glasses. (English) Zbl 1152.82011

Summary: Given numbers \(a_{ij} \geq 0\) for \(1 \leq i < j \leq N\), and given numbers \(b_i \geq 0\), \(i\leq N\), we consider the random Hamiltonian \(\sum_{i,j \leq N} \sqrt{a_{ij}} g_{ij} \sigma_i \sigma_j + \sum_{i \leq N} \sqrt{b_i} g_i \sigma_i\), where \(g_i , g_{ij}\) denote independent standard normal r.v., and where \(\sigma_i = \pm 1\). We give sufficient conditions on the coefficients \(a_{ij}\) for the system governed by this Hamiltonian to exhibit “high-temperature behavior”. There results extend known facts concerning the behavior of the Sherrington-Kirkpatrick model at “very high-temperature”. In a similar manner we give a general form of the “perceptron model”.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60G15 Gaussian processes
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References:

[1] Talagrand, M., Spin Glasses, a Challenge for Mathematicians (2003), Heidelberg: Springer, Heidelberg · Zbl 1033.82002
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