The \(k\)-tacnode process. (English) Zbl 1422.60137
Summary: 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 \(k\in \mathbb{Z}\), 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 Painlevé-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.
MSC:
| 60J65 | Brownian motion |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33E17 | Painlevé-type functions |
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