The k-tacnode process

TL;DR

This paper generalizes the tacnode process for Dyson Brownian bridges on the circle with drift, revealing new behaviors involving particles wrapping around the circle and deriving related probabilities and kernels.

Contribution

It introduces a generalized tacnode process for Dyson Brownian bridges with drift on the circle, including explicit formulas for winding probabilities and correlation kernels.

Findings

Derived winding number probabilities for the generalized process.

Obtained an explicit correlation kernel in terms of Painleve-II related functions.

Showed the process involves particles wrapping around the circle in the scaling limit.

Abstract

The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any integer k, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings-McLeod solutions to the inhomogeneous Painleve-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.