Airy processes with wanderers and new universality classes

TL;DR

This paper introduces new universality classes in nonintersecting Brownian bridges, revealing novel Airy and Pearcey processes influenced by wanderers, with implications for understanding complex stochastic systems near critical points.

Contribution

It develops a new Airy process with wanderers and explores the limiting behaviors leading to Pearcey processes and a novel kernel involving quintic polynomials.

Findings

New Airy process with wanderers near the edge of the spectrum

Emergence of Pearcey processes at cusp points with multiple wanderers

A potential new kernel involving a double integral with a quintic polynomial

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.