A Fuchsian matrix differential equation for Selberg correlation integrals

TL;DR

This paper derives a Fuchsian matrix differential equation to characterize Selberg correlation integrals, enabling explicit computation of power series expansions and asymptotics, with applications to singularity effects and polynomial zeros.

Contribution

It introduces a Fuchsian matrix differential equation framework for Selberg integrals and computes explicit connection matrices, advancing analytical tools for these integrals.

Findings

Explicit connection matrix between solution bases is calculated.

Power series for marginal distributions are derived.

Leading order asymptotics for small x are obtained in specific cases.

Abstract

We characterize averages of $\prod_{l=1}^N|x - t_l|^{\alpha - 1}$ with respect to the Selberg density, further contrained so that $t_l \in [0,x]$ $(l=1,...,q)$ and $t_l \in [x,1]$ $(l=q+1,...,N)$, in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the $t_j$ $(j=1,...,N)$. In the case $q=0$ and $\alpha < 1$ we compute the explicit leading order term in the $x \to 0$ asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case $q=0$ and $\alpha \in \mathbb Z^+$, and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.