A multi-dimensional Markov chain and the Meixner ensemble

TL;DR

This paper derives a determinant formula for the transition probabilities of a multi-dimensional Markov chain related to last-passage times, connecting it to the Meixner ensemble and providing a Fredholm determinant representation.

Contribution

It introduces a new determinant formula for the Markov chain's transition probabilities and links it to the Meixner ensemble with a Fredholm determinant representation.

Findings

Determinant formula for transition probabilities of the Markov chain

Connection to the Meixner ensemble for distribution functions

Fredholm determinant representation with double contour integral kernel

Abstract

We show that the transition probability of the Markoc chain $(G(j,1),...,G(j,n))_{j\ge 1}$, where the $G(i,j)'s$ are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for $G(m,n)$. We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.