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Polyanalytic Hermite polynomials associated with the elliptic Ginibre model

Nizar Demni, Zouhaïr Mouayn

TL;DR

The paper addresses constructing natural polyanalytic Hermite polynomials for the elliptic Ginibre model by combining the magnetic-Laplacian framework with a Bogoliubov-type squeezing of creation/annihilation operators. It derives explicit squeezed complex Hermite polynomials $H_{m,n}(z,\overline{z};\tau)$ that form an orthogonal basis of the $n$-th Landau eigenspace, relates them to $2D$-Hermite polynomials via a unimodular matrix $R_{\tau}$, and provides a concrete integral transform $T_{\tau,n}$ with kernel $W_{\tau,n}$ linking holomorphic and polyanalytic sectors. The work also gives a detailed kernel interpretation in terms of two-photon coherent states and the SU(1,1) metaplectic representation, enriching the probabilistic and quantum-mechanical perspectives on the elliptic Ginibre ensemble. These results yield new tools for studying determinantal processes in the elliptic regime and illuminate the geometric structure of polyanalytic bases in deformed Bargmann–Fock-type spaces.

Abstract

Motivated by the connection between the eigenvalues of the complex Ginibre matrix model and the magnetic Laplacian in the complex plane, we derive analogues of the complex Hermite polynomials for the elliptic Ginibre model. To this end, we appeal to squeezed creation and annihilation operators arising from the Bogoliubov transformation of creation and annihilation operators on the Bargmann-Fock space. The obtained polynomials are then expressed as linear combinations of products of Hermite polynomials and share the same orthogonality relation with holomorphic Hermite polynomials. Moreover, this expression allows to identify them with the 2D-Hermite polynomials associated to a unimodular complex symmetric 2x2 matrix. Afterwards, we derive, for any Landau level, a closed formula for the kernel of the isometry mapping the basis of (rescaled) holomorphic Hermite polynomials to the corresponding complex Hermite polynomials. This kernel is also interpreted in terms of the two-photon coherent states and the metaplectic representation of the SU(1,1) group.

Polyanalytic Hermite polynomials associated with the elliptic Ginibre model

TL;DR

The paper addresses constructing natural polyanalytic Hermite polynomials for the elliptic Ginibre model by combining the magnetic-Laplacian framework with a Bogoliubov-type squeezing of creation/annihilation operators. It derives explicit squeezed complex Hermite polynomials that form an orthogonal basis of the -th Landau eigenspace, relates them to -Hermite polynomials via a unimodular matrix , and provides a concrete integral transform with kernel linking holomorphic and polyanalytic sectors. The work also gives a detailed kernel interpretation in terms of two-photon coherent states and the SU(1,1) metaplectic representation, enriching the probabilistic and quantum-mechanical perspectives on the elliptic Ginibre ensemble. These results yield new tools for studying determinantal processes in the elliptic regime and illuminate the geometric structure of polyanalytic bases in deformed Bargmann–Fock-type spaces.

Abstract

Motivated by the connection between the eigenvalues of the complex Ginibre matrix model and the magnetic Laplacian in the complex plane, we derive analogues of the complex Hermite polynomials for the elliptic Ginibre model. To this end, we appeal to squeezed creation and annihilation operators arising from the Bogoliubov transformation of creation and annihilation operators on the Bargmann-Fock space. The obtained polynomials are then expressed as linear combinations of products of Hermite polynomials and share the same orthogonality relation with holomorphic Hermite polynomials. Moreover, this expression allows to identify them with the 2D-Hermite polynomials associated to a unimodular complex symmetric 2x2 matrix. Afterwards, we derive, for any Landau level, a closed formula for the kernel of the isometry mapping the basis of (rescaled) holomorphic Hermite polynomials to the corresponding complex Hermite polynomials. This kernel is also interpreted in terms of the two-photon coherent states and the metaplectic representation of the SU(1,1) group.

Paper Structure

This paper contains 12 sections, 137 equations.

Table of Contents

  1. Introduction and main results
  2. The Ginibre model and complex Hermite polynomials
  3. Deformation: the elliptic Ginibre model
  4. Squeezed complex Hermite polynomials
  5. The magnetic picture: $\tau =0$
  6. The deformed setting $\tau \in \left[ 0,1\right)$: squeezing the lowest Landau level
  7. Defining and computing the squeezed complex Hermite polynomials
  8. Polyanalytic Hermite polynomials associated with the elliptic Ginibre model
  9. Relation to $2D$-Hermite polynomials$:$ proof of Corollary 1.2
  10. An integral transform $T_{\tau ,m}:\mathcal{F}_{\tau }\left( \mathbb{C}\right) \rightarrow \mathcal{A}_{n}\left( \mathbb{C}\right)$
  11. The kernel $W_{\tau ,n}\left( z,w\right)$ of $T_{\tau ,m}$
  12. A quantum mechanical interpretation of the kernel $W_{\tau ,n}\left( z,w\right)$