Distribution of the first particle in discrete orthogonal polynomial ensembles

TL;DR

This paper presents a recurrence-based method to determine the distribution of the first particle in discrete orthogonal polynomial ensembles, connecting it to integrable systems and Painleve equations under rational weight conditions.

Contribution

It introduces a novel recurrence procedure for the distribution function, linking discrete orthogonal polynomial ensembles to integrable systems and Painleve equations.

Findings

Recurrence procedure computes the distribution function efficiently.

Connection established between orthogonal polynomial ensembles and Painleve equations.

Applicable to classical special cases with rational weight derivatives.

Abstract

We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational. In a number of classical special cases the recurrence procedure is equivalent to the difference and q-Painleve equations of chao-dyn/9507010, [Sakai]. Our approach is based on the formalism of discrete integrable operators and discrete Riemann--Hilbert problems developed in math.CO/9912093, math-ph/0111008.