Eigenvalue correlations on Hyperelliptic Riemann surfaces

TL;DR

This paper computes the functional derivative of the induced charge density on a union of intervals, linking it to eigenvalue fluctuations in Hermitian random matrix ensembles with specific support.

Contribution

It provides a novel calculation of the functional derivative of charge density on hyperelliptic Riemann surfaces, connecting it to eigenvalue fluctuations in random matrix theory.

Findings

Derived the functional derivative of charge density on hyperelliptic Riemann surfaces.

Connected the derivative to eigenvalue fluctuations in Hermitian random matrices.

Applicable to ensembles with eigenvalue support on multiple disjoint intervals.

Abstract

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.