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The extreme statistics of some noncolliding Brownian processes

Mustazee Rahman

Abstract

We consider certain noncolliding interacting particle systems driven by Brownian noise. A key example is drifted Brownian motions conditioned not to intersect and related models of eigenvalues of Hermitian random matrices. We establish limit theorems for the extremal particle. We find: (i) the scaling limit of the largest eigenvalue of Brownian motion over Hermitian, positive-definite matrices, (ii) Airy process limit for the largest eigenvalue of Dyson's Brownian motion for GUE started from generic initial conditions, and (iii) a Fredholm determinant formula for the running maximum of the top path among noncolliding Brownian bridges and, as a byproduct, a new formula for the law of largest eigenvalue in the Laguerre Orthogonal Ensemble as well as for a related point-to-line last passage percolation model.

The extreme statistics of some noncolliding Brownian processes

Abstract

We consider certain noncolliding interacting particle systems driven by Brownian noise. A key example is drifted Brownian motions conditioned not to intersect and related models of eigenvalues of Hermitian random matrices. We establish limit theorems for the extremal particle. We find: (i) the scaling limit of the largest eigenvalue of Brownian motion over Hermitian, positive-definite matrices, (ii) Airy process limit for the largest eigenvalue of Dyson's Brownian motion for GUE started from generic initial conditions, and (iii) a Fredholm determinant formula for the running maximum of the top path among noncolliding Brownian bridges and, as a byproduct, a new formula for the law of largest eigenvalue in the Laguerre Orthogonal Ensemble as well as for a related point-to-line last passage percolation model.

Paper Structure

This paper contains 19 sections, 19 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. The largest eigenvalue in a random matrix model
  3. Brownian motion over Hermitian positive-definite matrices
  4. Dyson's Brownian motion for GUE and universality of the Airy process
  5. The Airy process
  6. Universality
  7. Noncolliding Brownian bridges and point-to-line last passage percolation
  8. Noncolliding Brownian bridges
  9. Preliminaries
  10. Determinant formula for Brownian last passage percolation with drifts and a boundary
  11. Inhomogeneous Geometric last passage percolation
  12. Limit transition from Geometric to Brownian
  13. Proof of Theorem \ref{['thm:BLPP']}
  14. Limits of the constituent kernels
  15. Computation of some kernels: the narrow wedge and flat boundaries
  16. ...and 4 more sections

Key Result

Theorem 1

Consider, for each $n$, the model eqn:Hreg and assume the eigenvalues of $H^{\mathrm{reg}} = H^{\mathrm{reg}}_n$ are $\lambda^n_i = \lambda_1^n - \Delta \cdot (i-1)$ for $1 \leq i \leq n$ with a fixed $\Delta > 0$. Define the rescaled random variable Then, for each $a \in \mathbb{R}$, where the integral kernel $K_{\Delta}$ is as follows. Here $\gamma_{ver}$ is the vertical contour $\{\Re(z) = 1

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1.1
  • Corollary 1.1
  • Theorem 2
  • Theorem 3
  • Corollary 1.2
  • Theorem 4
  • Remark 3.1
  • Proposition 3.1
  • proof
  • ...and 21 more