Etudes in Integrable Probability 118 views

This thesis presents results from three papers in integrable probability, each exploiting a different structural property of an exactly solvable model. In the first part, we study the perturbed GUE corners process — the joint distribution of eigenvalues of all principal submatrices of a Gaussian Unitary Ensemble matrix perturbed by a fixed diagonal matrix. While the eigenvalue distribution at a fixed level is symmetric in the perturbation parameters, the full corners process is not. We construct explicit Markov swap operators that realize the transposition of adjacent parameters. When the parameters form an arithmetic progression, composing these operators yields a deterministic shift property for a system of reflected Brownian motions with drifts. In the second part, we study noncolliding q-exchangeable random walks on the nonnegative integers, where a q-exchangeable walk is one whose trajectory probability is multiplied by q whenever two neighboring increments are transposed. We construct a system of such noncolliding walks via a Doob h-transform and show that it forms a determinantal point process with an explicit double contour integral kernel. In an appropriate scaling limit, we establish a deterministic limit shape and prove that bulk local statistics converge to the incomplete beta kernel. In the third part, we study random permutations produced by the q-Demazure product — a probabilistic operation on the symmetric group that interpolates between the Demazure product and ordinary multiplication. Applied to a random subword of the longest permutation, it yields a random permutation depending on two parameters. We prove that the limiting permuton depends on these parameters only through a single combination, so the two-parameter model reduces to a one-parameter family. This resolves a conjecture of Morales, Panova, Petrov, and Yeliussizov on the limiting permuton.

PHD (Doctor of Philosophy)

Integrable Probability; random tilings; stochastic vertex models

English

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2026-05-01