The k-tacnode process

• 成果类型: Article
• 署名作者: Buckingham, Robert; Liechty, Karl
• 署名单位: DePaul University; University System of Ohio; University of Cincinnati
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0885-2
• 发表日期: 2019
• 页码: 341-395
• 关键词: nonintersecting brownian motions; large-degree asymptotics; critical-behavior; Orthogonal polynomials; UNIVERSALITY; equation; Respect; distributions; TRANSITION; EIGENVALUE

摘要

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 is an element of 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 Painleve-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.