Home /  Non-intersecting Brownian Motions at a Tacnode: Soft and Hard Edge Case.


Non-intersecting Brownian Motions at a Tacnode: Soft and Hard Edge Case. November 30, 2010
Parent Program: --
Location: SLMath: Eisenbud Auditorium
Description Speaker: Steven Delvaux

This talk deals with a model of $n$ non-intersecting Brownian motions with
two prescribed starting points at time $t=0$ and two prescribed ending
points at time $t=1$. We focus on the critical regime where the paths fill
two tangent ellipses in the time-space plane as $n \to \infty$. The
touching point of the two ellipses is called the \emph{tacnode}. The
limiting mean density for the positions of the Brownian paths at the time
of tangency consists of two touching semicircles, possibly of different
sizes. We show that in an appropriate double scaling limit, there is a new
family of limiting determinantal point processes with  integrable
correlation kernels that are expressed in terms of a new Riemann-Hilbert
problem of size 4x4. We describe a remarkable connection with the Hastings-McLeod solution of the
Painleve II equation. Finally we discuss a hard-edge variant of the
tacnode problem for non-intersecting squared Bessel paths.
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
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