Nonintersecting random walks in the neighborhood of a symmetric tacnode

• Autores: Mark Adler, Patrik L. Ferrari, Pierre van Moerbeke
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 4, 2013, págs. 2599-2647
• Idioma: inglés
• DOI: 10.1214/11-AOP726
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• Resumen

  • Consider a continuous time random walk in Z with independent and exponentially distributed jumps ±1. The model in this paper consists in an infinite number of such random walks starting from the complement of {−m,−m+1,…,m−1,m} at time −t, returning to the same starting positions at time t, and conditioned not to intersect. This yields a determinantal process, whose gap probabilities are given by the Fredholm determinant of a kernel. Thus this model consists of two groups of random walks, which are contained within two ellipses which, with the choice m≃2t to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling m=⌊2t+σt1/3⌋, where parameter σ controls the strength of interaction between the two groups of random walkers.