Randomness on the lattice

Abstract

In this lecture we review recent lattice QCD studies of the statistical properties of the eigenvalues of the QCD Dirac operator. We find that the fluctuations of the smallest Dirac eigenvalues are described by chiral Random Matrix Theories with the global symmetries of the QCD partition function. Deviations from chiral Random Matrix Theory beyond the Thouless energy can be understood analytically by means of partially quenched chiral perturbation theory.

A statistical analysis of hadron spectrum: Quantum chaos in hadrons

“…It would be interesting to see how quantum chaos emerges in hadrons as a result of quarkgluon underlying dynamics. Lattice QCD studies in this direction are underway [7,8,9].…”

“…One of the early successes of statistical analysis was Wigner's celebrated discovery of the fact that fluctuation properties of complex nuclear spectra are described by the Random Matrix Theory (RMT) [1,2,3,4,5]. It was later realized that these properties, referred to as Wigner-Dyson properties, have a great deal of universality and appear in spectra of many physical systems, ranging from quantum dots [6] to lattice gauge theories [7,8,9,10].…”

“…Recent theoretical work in lattice QCD [7,8,9] and quantum-mechanical Yang-Mills models [10,14] has already addressed the statistical properties of QCD spectrum as well as the quantum chaos aspects. Our findings will provide certain empirical support of these results and, hopefully, motivate further studies in this direction.…”