Date:
Thu, 24/06/202114:30-15:30
Broadcast: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=25147583-831c-4e92-b547-ad4a00755a9d
Title: Three ways to compute the free energy of pure spherical spin glasses
Abstract: Take a random homogeneous polynomial of degree p in N real variables x=(x_1,...,x_N) whose coefficients are independent Gaussian variables with variance 1/N. Call H(x) its restriction to the unit sphere in R^N. For any fixed level E>0, what is the volume of the level set of H(x) asymptotically as N tends to infinity? This question is, in fact, equivalent to a fundamental problem in mean-field spin glass theory of computing the free energy associated to H(x) --- which in statistical physics is called the pure p-spin spherical model.
In this colloquium seminar I will discuss three ways to compute the free energy of the spherical pure p-spin model. I will start with the famous formula proposed by G. Parisi in 1980 and proved by Talagrand more than two decades later (which works in a far more general setting). Next, I will describe a somewhat more direct approach to compute the free energy at low temperature which involves an analysis of critical points and the local behaviour around them. Finally, I will present a certain recent generalization of a representation for the free energy from the seminal 1975 paper of Thouless, Anderson and Palmer, and explain how it can be used to calculate the free energy.
The talk will assume no special knowledge in physics or probability.
Title: Three ways to compute the free energy of pure spherical spin glasses
Abstract: Take a random homogeneous polynomial of degree p in N real variables x=(x_1,...,x_N) whose coefficients are independent Gaussian variables with variance 1/N. Call H(x) its restriction to the unit sphere in R^N. For any fixed level E>0, what is the volume of the level set of H(x) asymptotically as N tends to infinity? This question is, in fact, equivalent to a fundamental problem in mean-field spin glass theory of computing the free energy associated to H(x) --- which in statistical physics is called the pure p-spin spherical model.
In this colloquium seminar I will discuss three ways to compute the free energy of the spherical pure p-spin model. I will start with the famous formula proposed by G. Parisi in 1980 and proved by Talagrand more than two decades later (which works in a far more general setting). Next, I will describe a somewhat more direct approach to compute the free energy at low temperature which involves an analysis of critical points and the local behaviour around them. Finally, I will present a certain recent generalization of a representation for the free energy from the seminal 1975 paper of Thouless, Anderson and Palmer, and explain how it can be used to calculate the free energy.
The talk will assume no special knowledge in physics or probability.