Airy processes with wanderers and new universality classes

Abstract

Consider $n+m$ nonintersecting Brownian bridges, with $n$ of them leaving from 0 at time $t=-1$ and returning to 0 at time $t=1$, while the $m$ remaining ones (wanderers) go from $m$ points $a_i$ to $m$ points $b_i$. First, we keep $m$ fixed and we scale $a_i,b_i$ appropriately with $n$. In the large-$n$ limit, we obtain a new Airy process with wanderers, in the neighborhood of $\sqrt{2n}$, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation. Letting the number $m$ of wanderers tend to infinity as well, leads to two Pearcey processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Upon tuning the starting and target points, one can let the two tips of the cusps grow very close; this leads to a new process, which might be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.

Published in at http://dx.doi.org/10.1214/09-AOP493 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Fluctuation properties of the TASEP with periodic initial configuration
• The Pearcey Process
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Differential Equations for Dyson Processes
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition
• Two speed TASEP

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