Local Statistics and Concentration for Non-intersecting Brownian Bridges With Smooth Boundary Data

TL;DR

This paper studies non-intersecting Brownian bridges with smooth boundary conditions, proving concentration bounds and showing their local statistics converge to the sine process, revealing universal behavior in the bulk.

Contribution

It establishes nearly optimal concentration bounds and proves convergence of local statistics to the sine process for non-intersecting Brownian bridges with smooth boundary data.

Findings

Nearly optimal concentration bounds for the bridges.

Convergence of bulk local statistics to the sine process.

Results hold under general boundary conditions without empty facets.

Abstract

In this paper we consider non-intersecting Brownian bridges, under fairly general upper and lower boundaries, and starting and ending data. Under the assumption that these boundary data induce a smooth limit shape (without empty facets), we establish two results. The first is a nearly optimal concentration bound for the Brownian bridges in this model. The second is that the bulk local statistics of these bridges along any fixed time converge to the sine process.