q -deformation of the Marchenko–Pastur law

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-λ /N, where N is the system size and λ ≥ 0, we derive the limiting spectral distribution as N→ ∞, which yields a q-deformation of the Marchenko–Pastur law. The limiting density undergoes a phase transition at an explicitly determined critical value λ c: for λ <λ c, the support consists of a single band region, whereas for λ>λ 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.