dorsal/arxiv
View Schema$q$-deformation of the Marchenko-Pastur law
| Authors | Sung-Soo Byun, Yeong-Gwang Jung, Guido Mazzuca |
|---|---|
| Categories | |
| ArXiv ID | 2601.09427vv1 |
| URL | https://arxiv.org/abs/2601.09427 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
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"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 $\\lambda\u003c\\lambda_c$, the support consists of a single band region, whereas for $\\lambda\u003e\\lambda_c$ 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.",
"arxiv_id": "2601.09427",
"authors": [
"Sung-Soo Byun",
"Yeong-Gwang Jung",
"Guido Mazzuca"
],
"categories": [
"math.PR",
"math-ph",
"math.CO",
"math.MP"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "$q$-deformation of the Marchenko-Pastur law",
"url": "https://arxiv.org/abs/2601.09427",
"version": "v1"
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