| In this work, we study invariant-class of random matrix ensembles characterized by the asymptotic logarithmic soft-confinement potential V(H) ∼ [lnH](1+lambda) (lambda > 0), named "lambda-ensembles". The suggestion is inspired by the existing random matrix models such as the critical ensembles (lambda=1), the free Lev´y matrices (lambda → 0 limit) and the Gaussian ensembles (lambda → infinity limit) in an effort to investigate the novel universality associated with the fat-tail random matrix ensembles as well as the logarithmic soft-confinement potential within the framework of rotationally invariant random matrix theory. First of all, we show that the orthogonal polynomials with respect to the weight function exp[-(ln x)1+lambda] belong to a novel orthogonal polynomial system, named "lambda-generalization of q-polynomials". Second, we show that based on numerical construction of the "lambda-generalization of q-polynomials", we can study the one-level and the two-level correlation functions as well as the level statistics of the lambda-ensembles. Third, we show that the one-level correlation (eigenvalue density) has a power-law form p(x) ∝ [ln x]lambda-1/ x and the unfolded two-level correlation function possesses the normal/anomalous structure, characteristic of the critical ensembles. We further show that the anomalous part, so-called "ghost-correlation peak" is controlled by the parameter lambda; decreasing lambda increases the anomaly. Third, we also identify the two-level kernel of the lambda-ensembles in the semi-classical 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. Forth, we show that the number variance is linear in L for all lambda and the slope (the level compressibility) is increasing as lambda decreases, which is consistent with the lambda-dependence of sum rule violation 0 < X(lambda) < 1. Finally, we will discuss the novel universality of the lambda-ensembles, which interpolates the Gaussian ensembles (lambda → infinity limit), the critical ensembles (lambda = 1), the free Levy matrices (lambda → 0 limit). |
