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The k-tacnode process

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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 \(k\in \mathbb {Z}\), 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.

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Authors and Affiliations

  1. DePaul University, Chicago, IL, USA

    Karl Liechty

  2. University of Cincinnati, Cincinnati, OH, USA

    Robert Buckingham

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  1. Robert Buckingham
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  2. Karl Liechty
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Correspondence to Karl Liechty.

Additional information

The authors thank the anonymous referee for helpful suggestions. They also thank Dong Wang, who was involved in this project at an early stage. Robert Buckingham was supported by by the National Science Foundation through Grant DMS-1615718 and by the Charles Phelps Taft Research Center through a Faculty Release Fellowship. Karl Liechty was supported by the Simons Foundation through Grant #357872.

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Buckingham, R., Liechty, K. The k-tacnode process. Probab. Theory Relat. Fields 175, 341–395 (2019). https://doi.org/10.1007/s00440-018-0885-2

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