Concentration of Measure Techniques and Applications

TL;DR

This paper explores various theories and applications of concentration of measure, including random operator compressions, eigenvector overlaps in Ginibre ensembles, and permutation asymptotics, advancing understanding in probability and random matrix theory.

Contribution

It generalizes existing approaches to concentration phenomena and introduces new methods for analyzing random permutations and matrix ensembles.

Findings

Concentration results for random operator compressions

Relation between matrix moments and eigenvector overlaps

Asymptotic analysis of Mallows permutations

Abstract

Concentration of measure is a phenomenon in which a random variable that depends in a smooth way on a large number of independent random variables is essentially constant. The random variable will "concentrate" around its median or expectation. In this work, we explore several theories and applications of concentration of measure. The results of the thesis are divided into three main parts. In the first part, we explore concentration of measure for several random operator compressions and for the length of the longest increasing subsequence of a random walk evolving under the asymmetric exclusion process, by generalizing an approach of Chatterjee and Ledoux. In the second part, we consider the mixed matrix moments of the complex Ginibre ensemble and relate them to the expected overlap functions of the eigenvectors as introduced by Chalker and Mehlig. In the third part, we develop a $q$-Stirling's formula and discuss a method for simulating a random permutation distributed according to the Mallows measure. We then apply the $q$-Stirling's formula to obtain asymptotics for a four square decomposition of points distributed in a square according to the Mallows measure. All of the results in the third part are preliminary steps toward bounding the fluctuations of the length of the longest increasing subsequence of a Mallows permutation.