# Janossy densities for chiral random matrix ensembles and their applications to two-color QCD

## TL;DR

This paper develops a numerical method to compute individual low-lying eigenvalue distributions of chiral random matrix ensembles and applies it to analyze the Dirac spectra in two-color QCD, enabling precise determination of the chiral condensate.

## Contribution

The paper introduces a compact formula for eigenvalue distributions and applies a Nyström-type quadrature method for accurate numerical evaluation in chiral ensembles.

## Key findings

- Numerical evaluation of eigenvalue distributions for chiral ensembles.
- Fitting of lattice QCD data to chiral symplectic ensemble predictions.
- Precise determination of the chiral condensate in two-color QCD.

## Abstract

We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nystr\"om-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nystr\"om-type method is obtained in an explicit form, and the $k^{\rm{\small th}}$ smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscopic limit. As an application of our result, the low-lying Dirac spectra of the SU(2) lattice gauge theory with $N_F=8$ staggered flavors are fitted to the numerical prediction from the chiral symplectic ensemble, leading to a precise determination of the chiral condensateof a two-color QCD-like system in the future.