[Paper Review] Scaling Limits of Planar Symplectic Ensembles

Gernot Akemann, Sung‐Soo Byun+1 more|arXiv (Cornell University)|Jun 17, 2021

Random Matrices and Applications26 references4 citations

TL;DR

This paper derives scaling limits of planar symplectic ensembles, focusing on the symplectic Ginibre ensemble with Gaussian and Mittag-Leffler-type potentials. It establishes a unifying kernel form at the edge near the real axis, connects bulk and edge behaviors, and derives Ward's equation for general potentials, proving universality and solving fractional differential equations for singular potentials using Mittag-Leffler functions.

ABSTRACT

We consider various asymptotic scaling limits $N o\infty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials $Q(ζ)=|ζ|^{2λ}-(2c/N)\log|ζ|$, with $λ>0$ and $c>-1$, the limiting kernel obeys a linear differential equation of fractional order $1/λ$ at the origin. For integer $m=1/λ$ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for planar symplectic ensembles for a general class of potentials. It serves as a tool to investigate the Gaussian and singular Mittag-Leffler universality class. This allows us to determine the functional form of all possible limiting kernels (if they exist) that are translation invariant, up to their integration domain.

Motivation & Objective

• To derive asymptotic scaling limits for the eigenvalue kernel of the symplectic Ginibre ensemble as N → ∞.
• To unify edge scaling near the real axis with bulk and complex Ginibre edge scaling via a single kernel form.
• To analyze singular potentials of Mittag-Leffler type with a logarithmic singularity at the origin.
• To derive and apply Ward’s equation for planar symplectic ensembles to characterize universal limiting kernels.
• To determine the functional form of translation-invariant limiting kernels up to their integration domain.

Proposed method

• Uses Pfaffian point process structure for integrability of the symplectic Ginibre ensemble.
• Applies asymptotic analysis to the kernel in the large-N limit for Gaussian and singular potentials.
• Derives a limiting kernel near the real axis edge using integral representations and Fourier transforms.
• Solves a fractional-order differential equation of order 1/λ at the origin for Mittag-Leffler potentials.
• Establishes Ward’s equation for general potentials by analyzing the functional form of the kernel.
• Uses the kernel’s structure and symmetry to prove universality and derive constraints on translation-invariant limits.

Experimental results

Research questions

• RQ1 How does the limiting kernel behave at the edge of the spectrum near the real axis for the symplectic Ginibre ensemble?
• RQ2 Can the edge kernel near the real axis unify with the bulk and complex Ginibre edge scaling?
• RQ3 What is the form of the limiting kernel for Mittag-Leffler potentials with a singularity at the origin?
• RQ4 How does Ward’s equation constrain the possible forms of translation-invariant limiting kernels?
• RQ5 Under what conditions does Ward’s equation hold for planar symplectic ensembles with general potentials?

Key findings

• The limiting kernel at the edge near the real axis is unified and connects to both bulk and complex Ginibre edge scaling.
• For Mittag-Leffler potentials with Q(ζ) = |ζ|^{2λ} − (2c/N)log|ζ|, the limiting kernel satisfies a fractional-order differential equation of order 1/λ.
• When 1/λ is an integer m, the solution is expressible in terms of Mittag-Leffler functions.
• Ward’s equation holds if and only if the support of the limiting kernel is connected (up to a null set).
• The functional form of all possible translation-invariant limiting kernels is determined up to their integration domain via Ward’s equation.
• The derivation confirms universality in the Gaussian and singular Mittag-Leffler classes, with explicit kernel structures.