arXiv subjects

Yeong-Gwang Jung

Publications and source records attributed to Yeong-Gwang Jung.

4 recordsLinked to original sources

arXiv2026

$q$-deformed polyanalytic Ginibre point processes:construction and central limit theorems for linear statistics

Based on the representation theory of the $q$-CCR algebra, we develop a representation-theoretic construction of the $q$-deformed polyanalytic Fock spaces, their reproducing kernels, and the associated $q$-deformed polyanalytic Bargmann transforms. The associated determinantal point processes provide $q$-deformations of the pure and full polyanalytic Ginibre ensembles. Moreover, we determine their limiting density and establish central limit theorems for the fluctuations of their linear statistics as the number of particles tends to infinity and the deformation parameter $q$ tends to 1 simultaneously. Remarkably, although the $q$-deformation changes the macroscopic density and the size of the droplet, the limiting covariance structures coincide, after spatial rescaling, with those of the classical polyanalytic Ginibre ensembles.

math.PR↗

arXiv2026

$q$-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 $λ\ge 0$, 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.

math.PR↗

arXiv2025

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight

We study $q$-deformed random unitary ensembles associated with the weight function of the Al-Salam--Carlitz orthogonal polynomials, indexed by a parameter $a < 0$. In the special case $a = -1$, the model reduces to the $q$-deformed Gaussian unitary ensemble. Employing the Flajolet--Viennot theory together with the combinatorics of matchings, we derive an explicit positive-sum expression for the spectral moments. In the double-scaling regime $q = e^{-λ/N}$, where $N$ denotes the ensemble size and $λ> 0$ is fixed, we derive the first two terms in the large-$N$ expansion of the spectral moments. As a consequence, we obtain a closed-form expression for the limiting spectral density. Notably, this density exhibits two successive phase transitions as $λ$ increases, characterised by a reduction in the number of soft edges from two, to one, and eventually to none. Furthermore, we show that the limiting density coincides with the limiting zero distribution of the Al-Salam--Carlitz orthogonal polynomials under the same scaling.

math-ph↗

arXiv2023

A universal framework for entanglement detection under group symmetry

One of the most fundamental questions in quantum information theory is PPT-entanglement of quantum states, which is an NP-hard problem in general. In this paper, however, we prove that all PPT $(\overlineπ_A\otimes π_B)$-invariant quantum states are separable if and only if all extremal unital positive $(π_B,π_A)$-covariant maps are decomposable where $π_A,π_B$ are unitary representations of a compact group and $π_A$ is irreducible. Moreover, an extremal unital positive $(π_B,π_A)$-covariant map $\mathcal{L}$ is decomposable if and only if $\mathcal{L}$ is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form $$Φ(ρ)=a\frac{\text{Tr}(ρ)}{d}\text{Id}_d+ bρ+cρ^T+(1-a-b-c)\text{diag}(ρ)$$ are entanglement-breaking, and that all A-BC PPT $(U\otimes \overline{U}\otimes U)$-invariant tripartite quantum states are A-BC separable. The former strengthens some conclusions in [VW01,KMS20], and the latter provides a strong contrast to the fact that there exist PPT-entangled $(U\otimes U\otimes U)$-invariant tripartite Werner states [EW01] and resolves some open questions raised in [COS18].

math-ph↗