arXiv subjects

Meg Walters

Publications and source records attributed to Meg Walters.

4 recordsLinked to original sources

arXiv2015

Concentration of Measure Techniques and Applications

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.

math.PR↗

arXiv2015

Phase Uniqueness for the Mallows Measure on Permutations

For a positive number $q$ the Mallows measure on the symmetric group is the probability measure on $S_n$ such that $P_{n,q}(\pi)$ is proportional to $q$-to-the-power-$\mathrm{inv}(\pi)$ where $\mathrm{inv}(\pi)$ equals the number of inversions: $\mathrm{inv}(\pi)$ equals the number of pairs $i \pi_j$. One may consider this as a mean-field model from statistical mechanics. The weak large deviation principle may replace the Gibbs variational principle for characterizing equilibrium measures. In this sense, we prove absence of phase transition, i.e., phase uniqueness.

math.PR↗

arXiv2014

A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble

We consider the mixed matrix moments for the complex Ginibre ensemble. These are well-known. We consider the relation to the expected overlap functions of Chalker and Mehlig. This leads to new asymptotic problems for the overlap. We obtain some results, but we also state some remaining open problems.

math-ph↗

arXiv2014

A method to derive concentration of measure bounds on Markov chains

We explore a method introduced by Chatterjee and Ledoux in a paper on eigenvalues of principle submatrices. The method provides a tool to prove concentration of measure in cases where there is a Markov chain meeting certain conditions, and where the spectral gap of the chain is known. We provide several additional applications of this method. These applications include results on operator compressions using the Kac walk on $SO(n)$ and a Kac walk coupled to a thermostat, and a concentration of measure result for the length of the longest increasing subsequence of a random walk distributed under the invariant measure for the asymmetric exclusion process.

math.PR↗