The influence of correlations on random energy models

Abstract

This thesis explores how correlations influence the extremes of variable-speed branching Brownian motion (VSBBM) and the limiting free energy of the continuous random energy model (CREM). These models are closely related and serve as toy models for mean field spin glass models such as the Sherrington-Kirkpatrick (SK) model.

In Chapter 1, we give a detailed introduction to VSBBM and the CREM. Also, we summarise the results of Chapter 2 and 3 and place them in the broader research context.

In Chapter 2, we study the extremes of a VSBBM where the time-dependent speed functions, which describe the time-inhomogeneous variance, converge to the identity function from below. We show that the log-correction for the order of the maximum depends only on the rate of convergence of the speed function near 0 and 1 and exhibits a smooth interpolation between the correction in the i.i.d. case, 1/(2 √2) ln t, and that of standard branching Brownian motion (BBM), 3/(2 √2) ln t. We prove that the limiting law of the maximum and the extremal process essentially coincide with those of standard BBM, using a first and second moment method which relies on the localisation of extremal particles.

In Chapter 3, we study the free energy of the CREM with the so-called Hamilton-Jacobi approach. This approach compares the limiting free energy of mean field spin glass models to the so-called viscosity solution of a Hamilton-Jacobi equation (HJE). For some of these models, this viscosity solution exactly matches the limiting free energy. In other cases such as the bipartite SK model, particularly when the so-called nonlinearity of the HJE is nonconvex, it is only proven that the viscosity solution provides a bound to the limiting free energy.

In the case of the CREM with a convex speed function, we establish a HJE on an infinite- dimensional space and prove that a unique viscosity solution exists, which is characterised by a variational formula. Also, we give an explicit description of the so-called initial condition of that HJE, which allows to simplify the variational formula of the viscosity solution. The limiting free energy of the CREM with a convex speed function is equal to the aforementioned viscosity solution.