Scaling Limits of Planar Symplectic Ensembles

Abstract

We consider various asymptotic scaling limits N → ∞ 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 a 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 a general class of potentials as a tool to investigate universality. This allows us to determine the functional form of kernels that are translation invariant up to its integration domain.

Universal scaling limits of the symplectic elliptic Ginibre ensemble

“…It is worth pointing out that the inhomogenous term in the second line of (1.6) corresponds to the (holomorphic) kernel of the complex elliptic Ginibre ensemble with dimension 2N . Such a relation has been observed in other models as well, which include the Laguerre [2,3,30] and the Mittag-Leffler ensembles [4,10,15].…”

“…One remarkable result in this direction was due to Akemann, Kieburg, Mielke, and Prosen [6], where the authors showed that away from the real axis, the eigenvalue statistics of the symplectic and complex Ginibre ensembles are equivalent in the large-N limit, see also [31] for a similar statement on the fluctuations of the maximal modulus. Moreover very recently, the edge scaling limit of the symplectic Ginibre ensemble at the right/left endpoint of the spectrum was obtained independently in [4] and [24,26]. These works can be thought of as the maximally non-Hermitian analogue of the previous work [7], which investigated the edge scaling limits of the elliptic Ginibre ensemble in the almost-Hermitian regime where 1 − τ = O(N − 1 3 ).…”

“…In our first result Proposition 1.1, we obtain a differential equation satisfied by the pre-kernel κ N , which can be recognised as a version of the Christoffel-Darboux formula, cf. [4,12,14,25].…”

“…This differential equation was utilized in [4]. See also [1] for a related statement in the other extremal case when τ = 1.…”

Random normal matrices in the almost-circular regime

“…In the spirit of [4,8,9], this may be used to study the existence of further universality classes beyond radially symmetric ensembles. We also refer to [31] and [2] for implementations of Ward's equations in the study of Hermitian random matrices and planar symplectic ensembles respectively. We remark that for a general class of external potentials, the local bulk and edge universality of random normal matrices were obtained in [6] and [29] respectively.…”