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

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $λ$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+λ)} \:(λ>0$). We showed that we can study eigenvalue correlations of these "$λ$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+λ}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $ρ(x) \propto [\ln x]^{λ-1}/x$ 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 $λ$; decreasing $λ$ increases the anomaly. We also identify the two-level kernel of the $λ$-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 $λ=1$. Finally, we discuss the universality of the $λ$-ensembles, which includes Wigner-Dyson universality ($λ\to \infty$ limit), the uncorrelated Poisson-like behavior ($λ\to 0$ limit), and a critical behavior for all the intermediate $λ$ ($0<λ<\infty$) 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.