Work

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

Year: 2026

Type: preprint

Abstract: 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.06904

Authors: Jung, Yeong-Gwang, Sato, Ryosuke

Institutions: Seoul National University, Hokkaido University

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Chuo University, National Research Foundation, National Research Foundation of Korea, Hokkaido University, Japan Society for the Promotion of Science

OpenAlex ID: W7212028558

Cites: 4

Cited by: 1

Related to: 10