A Fuchsian matrix differential equation for Selberg correlation integrals - Mathematical Physics

Abstract: We characterize averages of $\prod {l=1}^N|x - t l|^{\alpha - 1}$ withrespect 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 ofsolutions of a particular Fuchsian matrix differential equation. By making useof the Dotsenko-Fateev integrals, the explicit form of the connection matrixfrom the Frobenius type power series basis to this basis is calculated, thusallowing us to explicitly compute coefficients in the power series expansion ofthe averages. From these we are able to compute power series for the marginaldistributions of the $t j$ $j=1, .,N$. In the case $q=0$ and $\alpha < 1$ wecompute 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 dominatedstrong fluctuations. In the case $q=0$ and $\alpha \in \mathbb Z^+$, and withthe absolute values removed, the average is a polynomial, and we demonstratethat its zeros are highly structured.

Author: Peter J. Forrester, Eric M. Rains

Source: https://arxiv.org/