Spin glasses and random matrices
A first look at the critical temperature of a spin glass with random matrix theory, April 2025
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ChatGPT, quite hallucinatory. The exact prompt was “A first look at the critical temperature of a spin glass with random matrix theory”.
Synopsis
The Sherrington-Kirkpatrick model is the simplest example of a spin glass (Sherrington and Kirkpatrick 1975), where we just let the couplings of the Curie-Weiss model become Gaussian. The random interactions form a non-trivial behavior which is fertile ground for random matrix theory techniques. In this document we propose one classical computation of the critical temperature by Potters and Bouchaud (2020), with more theoretical context and explicit computations. This is an excuse to present the Harish-Chandra-Itzkynson- Zuber integral (Harish-Chandra 1957; Itzykson and Zuber 1980) in two forms taken from Potters and Bouchaud (2020) and Tao (2013).