Date of Award
Spring 2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Statistics and Data Science
First Advisor
Tatikonda, Sekhar
Abstract
This thesis studies structural properties of mean-field spin glasses in the replica-symmetric phase and establishes their equivalences in a general framework. We give a random projections result that characterizes the distribution of low-dimensional projections of random vectors exhibiting replica-symmetric properties of thin-shell and overlap concentration as approximately random Gaussians. A converse is provided, showing that asymptotically random Gaussian projections imply these hypotheses. This extends and unites several existing results in geometric functional analysis and spin glasses. We then present a new approach to local independence in spin glasses, i.e.~the phenomenon that any fixed subset of coordinates is asymptotically independent in the thermodynamic limit. The approach generalizes the rigorous cavity method from Talagrand by considering multiple cavity sites. Under replica-symmetric conditions, the random projections result kicks in and the cavity fields are revealed to be asymptotically independent, which in turn leads to local independence. Explicit expressions for the approximate marginals are given, which allow for a clean converse that shows local independence implies those replica-symmetric properties. The framework is general enough to encompass the classical and soft spin ($[-1,1]$) Sherrington-Kirkpatrick models, as well as the Gardner spin glasses, including the perceptron and Shcherbina-Tirozzi models. In the process, we present several decomposition theorems for the Gardner Hamiltonians that additively isolates their cavity fields, which are of standalone interest.
Recommended Citation
Wee, Lai Heng Timothy, "Characterizations of Replica-Symmetry in Mean-Field Spin Glasses" (2024). Yale Graduate School of Arts and Sciences Dissertations. 1394.
https://elischolar.library.yale.edu/gsas_dissertations/1394
