Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials

TL;DR

This paper explores a family of random matrix ensembles with logarithmic soft-confinement potentials, revealing their eigenvalue density, anomalous correlations, and universal behaviors across different parameter regimes, with implications for localization and delocalization phenomena.

Contribution

The study extends previous work by characterizing eigenvalue densities, two-level kernels, and universality classes of the $ ext{lambda}$-ensembles, including their critical and Poisson-like limits.

Findings

Eigenvalue density follows a power-law with logarithmic dependence.

Two-level kernel exhibits an anomalous, critical structure controlled by $ ext{lambda}$.

Universality spans from Wigner-Dyson to Poisson-like behaviors depending on $ ext{lambda}$.

Abstract

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $\lambda$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+\lambda)} \:(\lambda>0$). We showed that we can study eigenvalue correlations of these "$\lambda$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+\lambda}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $\rho(x) \propto [\ln x]^{\lambda-1}/x$ and ii) the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called "ghost-correlation peak", is controlled by the parameter $\lambda$; decreasing $\lambda$ increases the anomaly. We also identify the two-level kernel of the $\lambda$-ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for $\lambda=1$. Finally, we discuss the universality of the $\lambda$-ensembles, which includes Wigner-Dyson universality ($\lambda \to \infty$ limit), the uncorrelated Poisson-like behavior ($\lambda \to 0$ limit), and a critical behavior for all the intermediate $\lambda$ ($0<\lambda<\infty$) in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the $N$ dependence of the two-level kernel of the fat-tail random matrices.