Title: A graph-based equilibrium problem for the limiting distribution of non-intersecting Brownian motions at low temperature
Authors: Delvaux, Steven * ×
Kuijlaars, Arno * #
Issue Date: 2010
Publisher: Springer-Verlag New York
Series Title: Constructive Approximation vol:32 issue:3 pages:467-512
Abstract: We consider n nonintersecting 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)×(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.
ISSN: 0176-4276
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
* (joint) first author
× corresponding author
# (joint) last author

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