Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 007, 40 pages      arXiv:2106.09345

Scaling Limits of Planar Symplectic Ensembles

Gernot Akemann a, Sung-Soo Byun b and Nam-Gyu Kang b
a) Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany
b) School of Mathematics, Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

Received June 23, 2021, in final form January 19, 2022; Published online January 25, 2022

We consider various asymptotic scaling limits $N\to\infty$ 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 $Q(\zeta)=|\zeta|^{2\lambda}-(2c/N)\log|\zeta|$, with $\lambda>0$ and $c>-1$, the limiting kernel obeys a linear differential equation of fractional order $1/\lambda$ at the origin. For integer $m=1/\lambda$ 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.

Key words: symplectic random matrix ensemble; Pfaffian point process; Mittag-Leffler functions; Ward's equation; translation invariant kernel.

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