The influence of correlations on random energy models
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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.