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Absorption times for discrete Whittaker processes and non-intersecting Brownian bridges

Neil O'Connell

TL;DR

The paper investigates a conjectural link between absorption times $T_r$ of a discrete Whittaker process and the maximal heights of ensembles of non-intersecting Brownian bridges/excursions. It develops a representation-theoretic framework based on the quantum Toda chain and Whittaker functions, producing Karlin–McGregor type expansions for transition kernels and absorption times via left Mellin-transform eigenfunctions and right fundamental-Whittaker-coefficient eigenfunctions; explicit results are obtained for $r=1$ and $r=2$, with partial evidence for $r=3,4$ and higher rank. A key outcome is the conjectured correspondence $T_{2N}=2(M_N)^2$ and $T_{2N+1}=2(H_N)^2$ between absorption times and maximal heights of non-intersecting reflected Brownian bridges and excursions, respectively; asymptotically, $T_r$ is expected to converge to the GOE Tracy–Widom distribution under suitable centering and scaling. The work also yields a suite of factorisation identities, binomial-sum relations, and connections to discrete Gaussian ensembles, supporting the KPZ universality perspective and offering a framework for future exact and numerical investigations. Overall, the results blend integrable systems, representation theory, and probabilistic combinatorics to illuminate large-scale stochastic behavior in constrained Brownian systems.

Abstract

We present evidence for a conjectural relationship between absorption times for discrete Whittaker processes and maximal heights of non-intersecting Brownian bridges.

Absorption times for discrete Whittaker processes and non-intersecting Brownian bridges

TL;DR

The paper investigates a conjectural link between absorption times of a discrete Whittaker process and the maximal heights of ensembles of non-intersecting Brownian bridges/excursions. It develops a representation-theoretic framework based on the quantum Toda chain and Whittaker functions, producing Karlin–McGregor type expansions for transition kernels and absorption times via left Mellin-transform eigenfunctions and right fundamental-Whittaker-coefficient eigenfunctions; explicit results are obtained for and , with partial evidence for and higher rank. A key outcome is the conjectured correspondence and between absorption times and maximal heights of non-intersecting reflected Brownian bridges and excursions, respectively; asymptotically, is expected to converge to the GOE Tracy–Widom distribution under suitable centering and scaling. The work also yields a suite of factorisation identities, binomial-sum relations, and connections to discrete Gaussian ensembles, supporting the KPZ universality perspective and offering a framework for future exact and numerical investigations. Overall, the results blend integrable systems, representation theory, and probabilistic combinatorics to illuminate large-scale stochastic behavior in constrained Brownian systems.

Abstract

We present evidence for a conjectural relationship between absorption times for discrete Whittaker processes and maximal heights of non-intersecting Brownian bridges.
Paper Structure (16 sections, 4 theorems, 288 equations, 1 figure)

This paper contains 16 sections, 4 theorems, 288 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Eigenfunctions and their factorisations
  3. The Toda chain
  4. Class one Whittaker functions
  5. Fundamental Whittaker functions
  6. Mellin transforms of class one Whittaker functions
  7. Shifted coefficients of fundamental Whittaker functions
  8. Binomial sum identities
  9. Non-intersecting Brownian bridges
  10. Transition probabilities and absorption times
  11. The case $r=1$
  12. The general case
  13. The case $r=2$
  14. The case $r=3$
  15. The case $r=4$
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let Then

Figures (1)

  • Figure 1: Simulation of the Markov chain on $\Pi$, started with $\pi_{ij}=50$ for all $(i,j)$ and run until the first time that $\pi_{50,1}=0$. The height of this surface over the box with coordinates $(i,j)$ is the value of $\pi_{ij}$ at this stopping time, shown here for $1\le i,j\le 50$.

Theorems & Definitions (10)

  • Conjecture 1
  • Conjecture 2
  • Remark 2.1
  • Theorem 2.1: Stade
  • Theorem 2.2
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4