Nonintersecting Brownian bridges between reflecting or absorbing walls

TL;DR

This paper analyzes nonintersecting Brownian bridges with reflecting or absorbing walls, revealing new limiting kernels related to Pearcey and tacnode processes, and connects these kernels to Painlevé II equations.

Contribution

It introduces new hard-edge limiting kernels for nonintersecting Brownian bridges and links them to Painlevé II solutions, expanding understanding of boundary effects in such processes.

Findings

Derived hard-edge Pearcey and tacnode kernels for different wall conditions.

Established equivalence of single-time kernels to known Painlevé II-based kernels.

Constructed a Schlesinger transform preserving Painlevé II solutions.

Abstract

We study a model of nonintersecting Brownian bridges on an interval with either absorbing or reflecting walls at the boundaries, focusing on the point in space-time at which the particles meet the wall. These processes are determinantal, and in different scaling limits when the particles approach the reflecting (resp. absorbing) walls we obtain hard-edge limiting kernels which are the even (resp. odd) parts of the Pearcey and tacnode kernels. We also show that in the single time case, our hard-edge tacnode kernels are equivalent to the ones studied by Delvaux [16], defined in terms of a $4\times 4$ Lax pair for the inhomogeneous Painlev\'{e} II equation (PII). As a technical ingredient in the proof, we construct a Schlesinger transform for the $4 \times 4$ Lax pair in [16] which preserves the Hastings--McLeod solutions to PII.