Bipartite spherical spin glass at critical temperature (with a random matrix detour)

• Elizabeth Collins-Woodfin (McGill University)

One of the fascinating phenomena of spin glasses is the dramatic change in behavior that occurs between the high and low temperature regimes. The free energy of the spherical Sherrington-Kirkpatrick (SSK) model, for example, has Gaussian fluctuations at high temperature, but Tracy-Widom fluctuations at low temperature. A similar phenomenon holds for the bipartite SSK model, and we show that, when the temperature is within a small window around the critical temperature, the free energy fluctuations converge to an independent sum of Gaussian and Tracy-Widom random variables (joint work with Han Le). Our work follows two recent papers that proved similar results for the SSK model (by Landon and by Johnstone, Klochkov, Onatski, Pavlyshyn). Analyzing bipartite SSK at critical temperature requires a variety of tools including classical random matrix results, contour integral techniques, and a CLT for the log-characteristic polynomial of Laguerre (Wishart) random matrices evaluated near the spectral edge. This last ingredient was not present in the literature when we began our project, so I will discuss our proof of this CLT, which has other applications separate from bipartite spin glasses.

Asymptotic expansion of the Laplace functional for the Sine Process.

• Giuseppe Orsatti (UCLouvain-la-Neuve)

From the general theory of point process, the Laplace functional $\mathbb{E}[e^{-\int f d \xi}]$, where $f$ is in some specific class of functions , give us important information about the process itself. For Determinantal Point Process, the Laplace functional $\mathbb{E}[e^{-\lambda\int f d \xi}]$ can be written as a Fredholm determinant of an integral operator $\mathcal{K}$. In this talk we investigate the asymptotic expansion of $\mathbb{E}[e^{-\lambda\int f d \xi}]$ for the Sine process as $\lambda$ goes to $+\infty$. We demonstrate, via Riemann-Hilbert analysis, that the configurations $\lbrace\xi_j\rbrace_{j=1}^{\infty}$ in which the points are far from the global maximum of $f$ contribute more on the Laplace functional. This result will provide crucial information about the linear statistics and tail probabilities of the Sine process.

Discussions