Universality of the Pearcey process

Abstract

Consider non-intersecting Brownian motions on the line leaving from the origin and forced to two arbitrary points. Letting the number of Brownian particles tend to infinity, and upon rescaling, there is a point of bifurcation, where the support of the density of particles goes from one interval to two intervals. In this paper, we show that at that very point of bifurcation a cusp appears, near which the Brownian paths fluctuate like the Pearcey process. This is a universality result within this class of problems. Tracy and Widom obtained such a result in the symmetric case, when the two target points are symmetric with regard to the origin. This asymmetry enabled us to improve considerably a result concerning the non-linear partial differential equations governing the transition probabilities for the Pearcey process, obtained by Adler and van Moerbeke.

References in corpus (5)

• The Pearcey Process
• Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
• Differential Equations for Dyson Processes
• Lectures on random matrix models. The Riemann-Hilbert approach
• Gaussian matrix model in an external field and non-intersecting Brownian motions

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• From the Pearcey to the Airy process
• Asymptotics of Fredholm determinant associated with the Pearcey kernel
• Nonlinear PDEs for gap probabilities in random matrices and KP theory
• A periodic hexagon tiling model and non-Hermitian orthogonal polynomials
• Spectra of Random Hermitian Matrices with a Small-Rank External Source: The critical and near-critical regimes
• Random matrix minor processes related to percolation theory
• Pearcey universality at cusps of polygonal lozenge tiling
• Stochastic differential equations related to random matrix theory
• A PDE for Nonintersecting Brownian Motions and Applications