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A Graph-based Equilibrium Problem for the Limiting Distribution of Nonintersecting Brownian Motions at Low Temperature.
- Source :
-
Constructive Approximation . Dec2010, Vol. 32 Issue 3, p467-512. 46p. 1 Illustration, 7 Diagrams, 5 Graphs. - Publication Year :
- 2010
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01764276
- Volume :
- 32
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Constructive Approximation
- Publication Type :
- Academic Journal
- Accession number :
- 54002940
- Full Text :
- https://doi.org/10.1007/s00365-010-9106-7