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A Graph-based Equilibrium Problem for the Limiting Distribution of Nonintersecting Brownian Motions at Low Temperature

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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.

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Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Steven Delvaux & Arno B. J. Kuijlaars

Authors
  1. Steven Delvaux
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  2. Arno B. J. Kuijlaars
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Corresponding author

Correspondence to Steven Delvaux.

Additional information

Communicated by Percy A. Deift.

The first author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).

The work of the second author is supported by FWO-Flanders project G.0427.09, by K.U. Leuven research grant OT/08/33, by the Belgian Interuniversity Attraction Pole P06/02, by the European Science Foundation Program MISGAM, and by grant MTM2008-06689-C02-01 of the Spanish Ministry of Science and Innovation.

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Delvaux, S., Kuijlaars, A.B.J. A Graph-based Equilibrium Problem for the Limiting Distribution of Nonintersecting Brownian Motions at Low Temperature. Constr Approx 32, 467–512 (2010). https://doi.org/10.1007/s00365-010-9106-7

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  • DOI: https://doi.org/10.1007/s00365-010-9106-7

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