Three-Parametric Marcenko–Pastur Density

Abstract

The complex Wishart ensemble is the statistical ensemble of $M \times N$ complex random matrices with $M \geq N$ such that the real and imaginary parts of each element are given by independent standard normal variables. The Marcenko--Pastur (MP) density $\rho(x; r), x \geq 0$ describes the distribution for squares of the singular values of the random matrices in this ensemble in the scaling limit $N \to \infty$, $M \to \infty$ with a fixed rectangularity $r=N/M \in (0, 1]$. The dynamical extension of the squared-singular-value distribution is realized by the noncolliding squared Bessel process, and its hydrodynamic limit provides the two-parametric MP density $\rho(x; r, t)$ with time $t \geq 0$, whose initial distribution is $\delta(x)$. Recently, Blaizot, Nowak, and Warchol studied the time-dependent complex Wishart ensemble with an external source and introduced the three-parametric MP density $\rho(x; r, t, a)$ by analyzing the hydrodynamic limit of the process starting from $\delta(x-a), a > 0$. In the present paper, we give useful expressions for $\rho(x; r, t, a)$ and perform a systematic study of dynamic critical phenomena observed at the critical time $t_{\rm c}(a)=a$ when $r=1$. The universal behavior in the long-term limit $t \to \infty$ is also reported. It is expected that the present system having the three-parametric MP density provides a mean-field model for QCD showing spontaneous chiral symmetry breaking.