Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble

Abstract

We show that the squared maximal height of the top path among $N$ non-intersecting Brownian bridges starting and ending at the origin is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of K. Johansson's result that the supremum of the Airy${}_2$ process minus a parabola has the Tracy-Widom GOE distribution, and as such it provides an explanation for how this distribution arises in models belonging to the KPZ universality class with flat initial data. The result can be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier.

Expanded introduction to include additional motivation, minor mistakes corrected

References in corpus (12)

• Fluctuation properties of the TASEP with periodic initial configuration
• The one-dimensional KPZ equation and its universality class
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Exact distribution of the maximal height of p vicious walkers
• Differential Equations for Dyson Processes
• A determinantal formula for the GOE Tracy-Widom distribution
• Nonintersecting Brownian excursions
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• The One-dimensional KPZ Equation and the Airy Process
• Noncolliding Brownian Motion and Determinantal Processes
• Maximum distributions of bridges of noncolliding Brownian paths
• How flat is flat in random interface growth?

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• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Interacting diffusions on positive definite matrices
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Scaling limits of permutations avoiding long decreasing sequences
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions