Abstract
Given numbers a ij ≥ 0 for 1 ≤ i < j ≤ N, and given numbers b i ≥ 0, i ≤ N, we consider the random Hamiltonian \(\sum_{i,j \le N} \sqrt{a_{ij}} g_{ij} \sigma_i \sigma_j + \sum_{i \le N} \sqrt{b_i} g_i \sigma_i\) , where g i , g ij denote independent standard normal r.v., and where σ i = ± 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”.
References
Talagrand M. (2003). Spin Glasses, a Challenge for Mathematicians. Springer, Heidelberg
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Talagrand, M. A general form of certain mean field models for spin glasses. Probab. Theory Relat. Fields 143, 97–111 (2009). https://doi.org/10.1007/s00440-007-0121-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0121-y
Mathematics Subject Classifications (2000)
- Primary: 82B44
- Secondary: 60G15