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.

arXiv commentExpanded introduction to include additional motivation, minor mistakes corrected

References in corpus (33)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Discrete polynuclear growth and determinantal processes
• Fluctuation properties of the TASEP with periodic initial configuration
• The one-dimensional KPZ equation and its universality class
• Spatial correlations of the 1D KPZ surface on a flat substrate
• 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
• Renormalization fixed point of the KPZ universality class
• A determinantal formula for the GOE Tracy-Widom distribution
• Airy processes and variational problems
• Continuum statistics of the Airy2 process
• Local behavior and hitting probabilities of the Airy1 process
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Nonintersecting Brownian excursions
• Distribution of the time at which N vicious walkers reach their maximal height
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• The One-dimensional KPZ Equation and the Airy Process
• Noncolliding Brownian Motion and Determinantal Processes
• Endpoint distribution of directed polymers in 1+1 dimensions
• Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
• The Airy_1 process is not the limit of the largest eigenvalue in GOE matrix diffusion
• Random matrix central limit theorems for nonintersecting random walks
• Maximum distributions of bridges of noncolliding Brownian paths
• Non-colliding Brownian bridges and the asymmetric tacnode process
• Airy processes with wanderers and new universality classes
• Non-colliding Brownian Motions and the extended tacnode process
• How flat is flat in random interface growth?
• The height of watermelons with wall
• Multiplicative functionals on ensembles of non-intersecting paths
• The height and range of watermelons without wall

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• How flat is flat in random interface growth?
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• The hard-edge tacnode process for Brownian motion
• 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