Interlacing Patterns in Exclusion Processes and Random Matrices

Abstract

In the last decade, there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects that appear in the limit of large matrices also arise in the long-time limit for interacting particles and growth models. Examples of these are the famous Tracy-Widom distribution function and the Airy₂ process.<br /> The objectives of this thesis are threefold: First, we discuss known relations between random matrices and some models in the Kardar-Parisi-Zhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion processes. For these models, in the limit of large time <i>t</i>, universality of fluctuations has been previously obtained. We consider the convergence to the limiting distributions and determine the (non-universal) first order corrections, which turn out to be a non-random shift of order <i>t</i>^-1/3. Subtracting this deterministic correction, the convergence is then of order <i>t</i>^-2/3. We also determine the strength of asymmetry in the exclusion process for which the shift is zero and discuss to what extend the discreteness of the model has an effect on the fitting functions.<br /> Second, we focus on the Gaussian Unitary Ensemble and its relation to the totally asymmetric simple exclusion process and discuss the appearance of the Tracy-Widom distribution in the two models. For this, we consider extensions of these systems to triangular arrays of interlacing points, the so-called Gelfand-Tsetlin patterns. We show that the correlation functions for the eigenvalues of the matrix minors for complex Dyson's Brownian motion have, when restricted to increasing times and decreasing matrix dimensions, the same correlation kernel as in the extended interacting particle system under diffusion scaling limit. We also analyze the analogous question for a diffusion on complex sample covariance matrices.<br /> Finally, we consider the minor process of Hermitian matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on Gelfand-Tsetlin patterns that also appears in a generalization of Warren's process, in which Brownian motions have level-dependent drifts. We will also show that this process arises in a diffusion scaling limit from the interacting particle system on Gelfand-Tsetlin patterns with level-dependent jump rates.