# On determinantal point processes and random tilings with doubly periodic weights

**Time: **
Fri 2020-06-12 10.00

**Subject area: **
Mathematics

**Doctoral student: **
Tomas Berggren
, Matematik (Inst.)

**Opponent: **
Professor Neil O'Connell, University College Dublin, Dublin, Irland

**Supervisor: **
Associate Professor Maurice Duits, Matematik (Avd.)

## Abstract

This thesis is dedicated to asymptotic analysis of determinantal point processes originating from random matrix theory and random tiling models. Our main interest lies in random tilings of planar domains with doubly periodic weights.

Uniformly distributed random tiling models are known to be a very rich class of models where many interesting phenomena can be observed. These models have therefore been under investigation for many years and many aspects of the models are by now well understood. Random tiling models with doubly periodic weights are in fact an even richer class of models. However, these models are much more difficult to analyze and for a thorough study of their behavior new ideas are needed. This thesis increases the understanding of random tiling models with doubly periodic weights.

The thesis consists of three papers and two chapters; one introductory and background chapter and one chapter giving an overview of the papers.

Paper A deals with linear statistics of the thinned Circular Unitary Ensemble and the thinned sine process. The thinning creates a transition from the Circular Unitary Ensemble respectively sine process to the Poisson process. We study a part of these transitions in detail.

In Papers B and C we study random tiling models with doubly periodic weights. These two papers constitute the main contribution of this thesis.

In Paper B we give a general method how to analyze a large family of random tiling models. In particular, we provide a double integral formula for the correlation kernel in terms of a Wiener-Hopf factorization of an associated matrix-valued function. We also present a recursive method on how to construct the Wiener-Hopf factorization.

The method developed in Paper B is used in Paper C to analyze the 2×k-periodic Aztec diamond. More precisely, we derive the correlation kernel for the Aztec diamond of finite size and give a detailed description of the model as the size tends to infinity.