Localization and free energy asymptotics in disordered statistical mechanics and random growth models

TL;DR

This dissertation develops techniques to analyze the asymptotic behavior of disordered statistical mechanics and random growth models, focusing on localization phenomena and free energy fluctuations in infinite-volume limits.

Contribution

It introduces new methods for studying localization and free energy asymptotics in spin glasses, directed polymers, and percolation models, connecting physical phenomena to mathematical properties.

Findings

Quantified localization in low-temperature spin glasses and directed polymers.

Established connections between localization and free energy properties.

Analyzed fluctuation orders in first- and last-passage percolation models.

Abstract

This dissertation develops, for several families of statistical mechanical and random growth models, techniques for analyzing infinite-volume asymptotics. In the statistical mechanical setting, we focus on the low-temperature phases of spin glasses and directed polymers, wherein the ensembles exhibit localization which is physically phenomenological. We quantify this behavior in several ways and establish connections to properties of the limiting free energy. We also consider two popular zero-temperature polymer models, namely first- and last-passage percolation. For these random growth models, we investigate the order of fluctuations in their growth rates, which are analogous to free energy.