UVictoria Dynamics and Probability Seminar: Arnab Sen

We study a mean-field spin glass model whose coupling distribution has a power-law tail with exponent \alpha \in (0, 2). This is known as Levy spin glasses in literature. Though it is a fully connected model, many of its important characteristics are driven by the presence of strong bonds that have a sparse structure. In this sense, the Levy model sits between the widely studied Sherrington-Kirkpatrick model (with Gaussian couplings) and diluted spin glass models, which are more realistic but harder to understand. In this talk, I will report a number of rigorous results on the Levy model. For example, when 1< \alpha < 2, in the high-temperature regime, we obtain the limit and the fluctuation of the free energy. Also, we can determine the behaviors of the site and bond overlaps at the high temperature. Furthermore, we establish a variational formula of the limiting free energy that holds at any temperature. Interestingly, when 0< \alpha < 1, the effect of the strong bonds becomes more pronounced, which significantly changes the behavior of the model. For example, the free energy requires a different normalization (N^{1/\alpha} vs N), and its limit has a simple description via a Poisson point process at any temperature. 

This is a joint work with Wei-Kuo Chen and Heejune Kim.