三参数Marcenko-Pastur密度
Three-Parametric Marcenko-Pastur Density
摘要：
复Wishart系综是具有$M \geq N$的$M \times N$复随机矩阵的统计系综，使得每个元素的实部和虚部由独立的标准正态变量给出。Marcenko Pastur（MP）密度$ρ(x; r), x \geq 0$描述了在缩放极限$N \to \infty$、具有固定矩形度$r=N/M \in (0, 1]$的$M \to \infty$中该系综中随机矩阵的奇异值的平方的分布。通过非碰撞平方贝塞尔过程实现了平方奇异值分布的动力学延拓，其流体动力学极限给出了初始分布为$δ(x)$的双参数MP密度$ρ(x; r, t)$随时间的变化.最近，Blazot，Nowak和Warchol研究了含外源的含时复杂Wishart系综，并通过分析从$δ(x-a), a > 0$开始的过程的流体动力学极限，引入了三参数MP密度$ρ(x; r, t, a)$.在本文中，我们给出了有用的表达式$ρ(x; r, t, a)$和进行系统的研究，观察到的关键时刻$t_{\rm c}(a)=a$时，$r=1$动态临界现象。还报告了长期极限$t \to \infty$的普遍行为。我们希望，具有三参数MP密度的本系统提供了一个平均场模型，表现出自发的手征对称性破缺的QCD。
英文摘要：
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 $ρ(x; r), x \geq 0$ 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 $ρ(x; r, t)$ with time $t \geq 0$, whose initial distribution is $δ(x)$. Recently, Blaizot, Nowak, and Warchol studied the time-dependent complex Wishart ensemble with an external source and introduced the three-parametric MP density $ρ(x; r, t, a)$ by analyzing the hydrodynamic limit of the process starting from $δ(x-a), a > 0$. In the present paper, we give useful expressions for $ρ(x; r, t, a)$ 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.
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