Free Energy Fluctuations of the Bipartite Spherical SK Model at Critical Temperature

The spherical Sherrington–Kirkpatrick (SSK) model and its bipartite analog both exhibit the phenomenon that their free energy fluctuations are asymptotically Gaussian at high temperature but asymptotically Tracy–Widom at low temperature. This was proved in two papers by Baik and Lee, for all non-critical temperatures. The case of the critical temperature was recently computed for the SSK model in two separate papers, one by Landon and the other by Johnstone, Klochkov, Onatski, Pavlyshyn. In the current paper, we derive the critical temperature result for the bipartite SSK model. In particular, we find that the free energy fluctuations exhibit a transition when the temperature is in a window of size \(n^{-1/3}\sqrt{\log n}\) $n^{-1/3}\sqrt{logn}$ around the critical temperature, the same window as for the SSK model. Within this transitional window, the asymptotic fluctuations of the free energy are the sum of independent Gaussian and Tracy–Widom random variables.