$q$-deformation of the Marchenko-Pastur law

TL;DR

This paper introduces a $q$-deformed version of the Marchenko-Pastur law, analyzing spectral distributions of a $q$-deformed ensemble with phase transitions, using multiple mathematical approaches to establish convergence and moments.

Contribution

It provides the first detailed analysis of a $q$-deformed Marchenko-Pastur law, including phase transition behavior and explicit spectral moment formulas.

Findings

Spectral distribution exhibits a phase transition at a critical $\lambda_c$.

Support of the distribution changes from single band to multiple regions.

Explicit formulas for spectral moments and large-$N$ expansions.

Abstract

We study a $q$-deformed random unitary ensemble associated with the little-$q$ Laguerre weight, which provides a discrete analogue of the classical Laguerre unitary ensemble. In the double scaling regime $q=e^{-\lambda/N}$, where $N$ is the system size and $\lambda \ge 0$, we derive the limiting spectral distribution as $N\to \infty$, which yields a $q$-deformation of the Marchenko-Pastur law. The limiting density undergoes a phase transition at an explicitly determined critical value $\lambda_c$: for $\lambda<\lambda_c$, the support consists of a single band region, whereas for $\lambda>\lambda_c$ an additional saturated region emerges adjacent to the band region. Our derivation of the limiting distribution is based on three complementary approaches: the method of moments, the analysis of a constrained equilibrium problem, and the asymptotic zero distribution of orthogonal polynomials. As a consequence, we establish the convergence of the empirical measure as well as a large deviation principle. In addition, we derive closed-form expressions for the spectral moments using the combinatorial structure of orthogonal polynomials, and obtain large-$N$ expansions for these moments.