The Spherical Sherrington-Kirkpatrick (SSK) model is defined by the Gibbs measure on a highdimensional sphere with a random Hamiltonian given by a symmetric quadratic function. The free energy at the zero temperature is the same as the largest eigenvalue of the random matrix associated with the quadratic function. Even for the finite temperature, there is a simple relationship between the free energy and the eigenvalues. We will discuss how one can study the fluctuations of the free energy using this relationship and results from random matrix theory. We will also discuss the distribution of the spin sampled from the Gibbs measure.[-]

The Gibbs measure of many disordered systems at low temperature may exhibit a very strong dependance on even tiny variations of temperature, usually called “temperature chaos”. I will discuss this question for Spin Glasses. I will report on a recent work with Eliran Subag (Courant) and Ofer Zeitouni (Weizmann and Courant), where we give a detailed geometric description of the Gibbs measure at low temperature, which in particular implies temperature chaos for a general class of spherical Spin Glasses at low temperature. This question has a very long past in the physics literature, and an interesting recent history in mathematics. Indeed, in 2015, Eliran Subag has given a very sharp description of the Gibbs measure for pure p-spin spherical Spin Glasses at low temperature, building on results on the complexity of these spin glasses by Auffinger-Cerny and myself. This description (close to the so-called Thouless-Anderson-Palmer picture) excludes the existence of temperature chaos for the pure p-spin!! The recent work gives an extension of this very detailed geometric description of the Gibbs measure to the case of general mixed models, and shows that in fact the pure p-spin is very singular.[-]