Date of Award

2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Jian Ding

Second Advisor

Robin Pemantle

Abstract

This thesis considers asymptotic behaviors of high-dimensional disordered systems, including Ising model and mean-field spin glass models. We first study the decay rate of correlations in the two-dimensional random field Ising model (RFIM). Second, we study the limit free energy of disordered systems.

For RFIM, we are interested in the two-dimensional case where the external field is of i.i.d centered Gaussian variables. We show that under nonnegative temperature, the effect of boundary conditions on the magnetization in a finite box decays exponentially in the side length of the box.

On the side of mean-field models, we use the Hamilton-Jacobi equation (HJE) approach, initiated by Jean-Christophe Mourrat, to characterize limiting free energy in many models from statistical inference problems and mean-field spin glass models. We now investigate infinite-dimensional models including many spin glass models and inference problems where the rank of the signal matrix increases as $n$ is sent to infinity. We give an intrinsic meaning to the Hamilton--Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness.This will shed more light on the mysterious Parisi formula as the limit of free energy in the Sherrington--Kirkpatrick model.

Included in

Mathematics Commons

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