Abstract

We consider the Sherrington--Kirkpatrick spin glass model and its mixed-p spin generalizations. The ground state energy is the maximum of the Hamiltonian, or the free energy when the temperature is set to be zero. The limiting ground state energy is known to be given by a Parisi-type formula. We establish the (one-sided) large deviation of this quantity by computing explicitly the limiting cumulant generating function. We also obtain an explicit variational formula for the rate function. There is also recent interest in understanding the algorithmic aspect of the ground state energy, since it is a typical optimization problem in a complex landscape. In particular, we obtain a formula for the ground state energy that is similar to the algorithmic threshold for Lipschitz algorithms. This is based on a joint work in preparation with Alice Guionnet, Justin Ko, Bertrand Lacroix-A-Chez-Toine, and Jean-Christophe Mourrat.