Distribution of the k-th smallest Dirac operator eigenvalue : an update

TL;DR

This paper introduces an efficient method based on random matrix theory and Fredholm determinants to evaluate the distribution of the k-th smallest Dirac eigenvalue in QCD, overcoming previous computational challenges.

Contribution

It presents a novel approach using Nystrom discretization to compute eigenvalue distributions, extending applicability to various cases and related physical systems.

Findings

Efficient computation of eigenvalue distributions in QCD and related theories.

Application to level spacings in Anderson Hamiltonian and high-temperature QCD.

Partial lifting of previous technical restrictions on flavor and topological charge.

Abstract

Based on the exact relationship to random matrix theory, we present an alternative method of evaluating the probability distribution of the k-th smallest Dirac eigenvalue in the epsilon-regime of QCD and QCD-like theories. By utilizing the Nystrom-type discretization of Fredholm determinants and Pfaffians, practical trouble of evaluating multiple integrations is circumvented and technical restrictions on the parities of the number of flavors and of the topological charge present in our previous treatment for beta=1 and 4 cases [Phys. Rev. D 63, 045012 (2001)] are partly lifted. This method is also applied to the distributions of spacings between k-th nearest-neighboring levels in the mobility edges of Anderson Hamiltonian and Dirac operator in high-temperature QCD.