Bordered and Framed Toeplitz and Hankel Determinants with Applications to Integrable Probability

TL;DR

This paper reviews bordered and framed Toeplitz and Hankel determinants, explores their connections to orthogonal polynomials, and applies these results to models in integrable probability, including non-intersecting paths and the six-vertex model.

Contribution

It provides a comprehensive review of structured determinants and introduces new asymptotic formulas for non-intersection probabilities in continuous-time random walks.

Findings

Derived asymptotic formulas for non-intersection probabilities.

Connected structured determinants to orthogonal polynomials and integrable models.

Applied results to non-intersecting path ensembles and the six-vertex model.

Abstract

Bordered and framed Toeplitz/Hankel determinants have the same structure as Toeplitz/Hankel determinants except in small number of matrix rows and/or columns. We review these structured determinants and their connections to orthogonal polynomials, collecting well-known and perhaps less well-known results. We present some applications for these structured determinants to ensembles of non-intersecting paths and the six-vertex model, with an eye towards asymptotic analysis. We also prove some asymptotic formulae for the probability of non-intersection for an ensemble of continuous time random walks for certain choices of starting and ending points as the number of random walkers tends to infinity.