2009/07/14 by Steven Delvaux, Arno B. J. Kuijlaars, Delvaux, Steven +1
Economics, Econometrics and Finance · Mathematics · #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.CV #math.PR #msc:31C99 #msc:35Q15
paper · pdf · doi:10.48550/arxiv.0907.2310
41 pages, 13 figures
arxiv created 2009/07/14 · arxiv updated 2009/12/01
We consider n non-intersecting Brownian motion paths with p prescribed starting positions at time t=0 and q prescribed ending positions at time t=1. The positions of the paths at any intermediate time are a determinantal point process, which in the case p=1 is equivalent to the eigenvalue distribution of a random matrix from the Gaussian unitary ensemble with external source. For general p and q, we show that if a temperature parameter is sufficiently small, then the distribution of the Brownian paths is characterized in the large n limit by a vector equilibrium problem with an interaction matrix that is based on a bipartite planar graph. Our proof is based on a steepest descent analysis of an associated (p+q) by (p+q) matrix valued Riemann-Hilbert problem whose solution is built out of multiple orthogonal polynomials. A new feature of the steepest descent analysis is a systematic opening of a large number of global lenses.