Taiki Endo | Searcharxiv

arXiv2022

Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases

We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the $R$-transforms and the $S$-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the $R$-transforms and the $S$-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.

math.PR↗

arXiv2020

Three-Parametric Marcenko-Pastur Density

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.

math.PR↗