$q$-deformation of the Marchenko-Pastur law

TL;DR

The paper introduces a $q$-deformation of the Marchenko–Pastur law through the little $q$-Laguerre unitary ensemble on a one-sided $q$-lattice with double-scaling $q=e^{-\lambda/N}$. It proves that the limiting spectral distribution is a $q$-deformed MP law $\rho^{(c)}(x)$ that exhibits a phase transition at $\lambda_c$, separating a single band from a band-plus-saturated region, and provides three complementary derivations via moments, constrained equilibrium, and zero distributions to establish convergence of the empirical measure and a large deviation principle. The authors also derive closed-form spectral moments and their large-$N$ expansions, tying discrete log-gas ensembles to integrable probability and last-passage percolation through the combinatorics of $q$-Laguerre polynomials. This work extends fundamental spectral laws of random matrix theory to $q$-deformed, discrete settings and offers rigorous tools for analyzing phase separation and large deviations in such models. The results deepen the connection between $q$-orthogonal polynomials, logarithmic potential theory under an upper constraint, and the asymptotics of discrete random matrix ensembles.

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 $q=e^{-λ/N}$, where $N$ is the system size and $λ\ge 0$, 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 $λ_c$: 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.

Figures (8)

• Figure 1: (A): The graph of $\lambda \mapsto \mathcal{M}_{p}^{\rm (qL)}$ (black solid) together with $\lambda \mapsto \frac{1}{N} m_{N,p}^{\rm (qL)}$ for $N=15$ (red, dotted), $N=30$ (blue, dot--dashed), and $N=60$ (purple, dashed). (B): The corresponding plot of $\lambda \mapsto m_{N,p}^{\rm (qL)} - N \mathcal{M}_{p}^{\rm (qL)}$ and its comparison with $\lambda \mapsto \mathcal{M}_{p,1/2}^{\rm (qL)}(d)$. Here $p=2$.
• Figure 2: The plots illustrate the density \ref{['eqn:def of limiting density']} for $c=0$ (first row) and $c=2$ (second row). The corresponding critical values are $\lambda_c\vert_{c=0}=\log 2$ and $\lambda_c\vert_{c=2}\approx 0.6823$. For $c=0$, the parameters are $\lambda=\log(4/3)<\lambda_c$ (left) and $\lambda=\log 3>\lambda_c$ (right). For $c=2$, the parameters are $\lambda=\log(4/3)<\lambda_c$ (left) and $\lambda=\log 2>\lambda_c$ (right). In addition, histograms of the zeros of the $q$-Laguerre polynomials for $N=4000$ are superimposed.
• Figure 3: A Motzkin path from $(0,3)$ to $(12,1)$ with weight $b_{1}b_{2}^{2}b_{3}\lambda_{2}^{2}\lambda_{3}^{2}\lambda_{4}$
• Figure 4: A bipartite matching in $M_{5,3}^{(2)}$. The dashed vertical line separates negative and positive vertices. No edge connects a vertex in $T_{-}$ to a vertex in $B_{-}$.
• Figure 5: A Laguerre history from $(0,3)$ to $(5,3)$ and its corresponding matching in $M_{5,3}^{(2)}$

Theorems & Definitions (44)

• Theorem 1.1: Spectral moments of the $q$-Laguerre unitary ensemble
• Theorem 1.2: $q$-deformed Marchenko-Pastur law via three approaches
• Remark 1: Saturated and band region
• Remark 2: Jackson $q$-integral
• Remark 3: The special case $c=0$: hard edge at the origin and $q$-deformed semi-circle law
• Remark 4: Continuum limit $q\to1$
• Remark 5: Large-$N$ expansion of spectral moments
• Remark 6: Upper constraint condition
• Remark 7: Zero distributions of orthogonal polynomials
• Remark 8: $\beta$-ensemble generalisation