The hard-edge tacnode process for Brownian motion

TL;DR

This paper studies the limiting behavior of non-intersecting Brownian bridges constrained below a threshold, revealing a new one-parameter family of processes with connections to statistical models like the six-vertex model and Aztec diamond.

Contribution

It introduces the hard-edge tacnode process as a new scaling limit for Brownian bridges conditioned on a boundary, extending understanding of constrained stochastic systems.

Findings

Derived the limiting distribution of the top Brownian bridge under constraints

Established the correlation kernel for the limiting process

Connected the process to models like the six-vertex and Aztec diamond

Abstract

We consider $N$ non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large $N$ limit, we determine the limiting distribution of the top Brownian bridge conditioned to stay below a function as well as the limiting correlation kernel of the system. It is a one-parameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the six-vertex model and to the Aztec diamond on restricted domains.