Non-intersecting squared Bessel paths at a hard-edge tacnode

TL;DR

This paper analyzes the critical behavior of non-intersecting squared Bessel paths at a hard-edge tacnode, revealing a new Riemann-Hilbert problem and a novel connection to the Painlevé II equation.

Contribution

It introduces a new Riemann-Hilbert problem of size 4x4 and a Lax pair for the Painlevé II equation, extending previous work to the inhomogeneous case with parameter .

Findings

Derived a new critical correlation kernel for the process.

Established a connection between the kernel and a Riemann-Hilbert problem.

Extended previous homogeneous case results to inhomogeneous parameters.

Abstract

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of $n$ non-intersecting squared Bessel paths, with all paths starting at the same point $a>0$ at time $t=0$ and ending at the same point $b>0$ at time $t=1$. Our interest lies in the critical regime $ab=1/4$, for which the paths are tangent to the hard edge at the origin at a critical time $t^*\in (0,1)$. The critical behavior of the paths for $n\to\infty$ is studied in a scaling limit with time $t=t^*+O(n^{-1/3})$ and temperature $T=1+O(n^{-2/3})$. This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size $4\times 4$. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlev\'e II equation $q"(x) = xq(x)+2q^3(x)-\nu,$ where $\nu=\alpha+1/2$ with $\alpha>-1$ the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang \cite{DKZ} for the homogeneous case $\nu = 0$.