Airy processes with wanderers and new universality classes (Q964782)

The authors 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 \(b_i\). When \(m=0\) and \(n\) becomes very large, the Airy process is known to describe the cloud of particles viewed from any point on the curve \({\mathcal C}:x=\sqrt{2n(1-t^2)}\), with time and space properly rescaled. In the present paper, the authors first keep \(m>0\) fixed and study the statistical fluctuations of the edge of the cloud of particles near any point on the curve \({\mathcal C}\). In the large-\(n\) limit, they obtain a new Airy process with wanderers. This process is governed by an Airy-type kernel, with a rational perturbation. Letting \(m\) tend to infinity as well, they consider \(m\) wanderers starting from the same point \(\tilde{a}=\alpha m^{1/3}\) and ending up at the same point \(\tilde{b}=\beta m^{1/3}\). This leads to two Pearcy processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Moreover, upon tuning the starting and target points, the two tips of the cusps can be let growing very close. This leads to a new process, which is conjectured to be governed by a kernel represented as a double integral involving the exponential of a quintic polynomial in the integration variables.