INS seminars 48: The tacnode kernel in non-intersecting Brownian motions and its transition (Lun Zhang, Jan.02, 2013)

INS seminars 48

Abstract:

Point processes with determinantal correlation kernels have attracted a lot of interest over the past few decades due to the rich mathematical structures behind them and their frequent occurrences in various random models including random matrix theory, random growth and tiling problems. In this talk, we consider a model of n one-dimensional non-intersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time-space plane as n tends to infinity. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles. We show that the local path correlations around this point are characterized by a family of limiting determinantal point processes with new integrable correlation kernels, namely, tacnode kernels, that are expressed in terms of new Riemann-Hilbert problem of size 4*4. Furthermore, we retrieve a canonical kernel -- the Pearcey kernel from the tacnode kernel in a certain double scaling limit. This leads to descriptions of phase transitions among different processes. This talk is based on joint work with Steven Delvaux, Dries Geudens and Arno Kuijlaars.