# Random matrix model approach to chiral symmetry

## TL;DR

This paper reviews how random matrix theory models can describe chiral symmetry in QCD, providing exact results for spectral properties and insights into phase transitions at finite temperature and chemical potential.

## Contribution

It introduces a chiral random matrix model aligned with QCD symmetries and derives universal spectral properties, extending to finite temperature and chemical potential scenarios.

## Key findings

- Exact universal properties of the Dirac spectrum are obtained.
- Variance of level counts is suppressed logarithmically.
- The model offers insights into the chiral phase transition and quenched approximation.

## Abstract

We review the application of random matrix theory (RMT) to chiral symmetry in QCD. Starting from the general philosophy of RMT we introduce a chiral random matrix model with the global symmetries of QCD. Exact results are obtained for universal properties of the Dirac spectrum: i) finite volume corrections to valence quark mass dependence of the chiral condensate, and ii) microscopic fluctuations of Dirac spectra. Comparisons with lattice QCD simulations are made. Most notably, the variance of the number of levels in an interval containing $n$ levels on average is suppressed by a factor $(\log n)/\pi^2 n$. An extension of the random matrix model model to nonzero temperatures and chemical potential provides us with a schematic model of the chiral phase transition. In particular, this elucidates the nature of the quenched approximation at nonzero chemical potential.