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

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 $t^{\ast }\in (0,1)$. The critical behavior of the paths for $n\to \infty$ is studied in a scaling limit with time $t=t^{\ast }+O(n^{-1/3})$ 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 Painlevé 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 $\nu =0$.

54 pages, 13 figures. Corrected error in Theorem 2.4

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Cited by in corpus (4)

• Nonintersecting Brownian motions on the unit circle
• The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
• The hard-edge tacnode process for Brownian motion
• Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains