Asymptotic geometry of discrete interlaced patterns: Part I

TL;DR

This paper analyzes the asymptotic geometric structure of discrete Gelfand-Tsetlin patterns, revealing their boundary shapes and local properties, and establishes a determinantal correlation kernel for these configurations.

Contribution

It introduces a determinantal structure for discrete Gelfand-Tsetlin patterns and characterizes their asymptotic boundary shapes and local geometric properties.

Findings

Established a determinantal correlation kernel.

Parameterised the asymptotic boundary shapes.

Analyzed local geometric properties of the boundary.

Abstract

A discrete Gelfand-Tsetlin pattern is a configuration of particles in Z^2. The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc, and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behaviour of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic `shape' of such systems. We provide parameterisations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behaviour of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the upcoming paper, [4], in which we resolve some of the remaining issues. Both of these papers serve as background material for the upcoming papers, [5] and [6], in which we examine the edge asymptotic behaviour.