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 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.

Truncations of random symplectic unitary matrices

“…Our asymptotic analysis of the pre-kernel g N (z, z ) in the regime of strong non-unitarity is carried out by extending the summation over k in (14) to N = ∞ (justified in Lemma 6.1) and employing an integral representation for the extended sum (Lemma 6.3) for the evaluation of this sum via the saddle point method. This approach is different to the ones used previously in the literature on quaternion-real ensembles and is well suited for the asymptotic analysis of the correlation functions in the complex bulk 3 . Near the real line it leads to a saddle point analysis which is hard to perform.…”

“…xy − (xy) N − N (xy) N (1 − xy) (1 − xy)3 .Recalling (108) and (109) and collecting all terms of order N 3 ,∂ 2 ∂x∂y A N (x, y) = O(N 2 ) + e iφ 0 1 − e 2iφ 0 2N 3 (2t) 3 −1 + e −2t 1 + 2t + 2t 2 .…”

“…This approach also works for other Gaussian matrix distributions, e.g., the elliptic deformation of the Ginibre ensemble [13], and, locally at the origin, for its chiral partner [4]. At the origin it also works for some non-Gaussian distributions, see [3] and references therein.…”

“…One strategy for calculating its scaling limits is to write a differential equation for the kernel suitable for asymptotic analysis. Such an approach was first used in [50] and then in different forms in [35] and [39] and yielded the eigenvalue correlation functions in the quaternion-real Ginibre ensemble locally at the origin and, after additional analysis in complex bulk [6], and also near the real line including the real edges [3,13]. This approach also works for other Gaussian matrix distributions, e.g., the elliptic deformation of the Ginibre ensemble [13], and, locally at the origin, for its chiral partner [4].…”

Universality of the number variance in rotational invariant two-dimensional Coulomb gases

“…which is suitable according to Definition 1.1. Such a model was studied in [19,21] for β = 2 and [4] for β = 4. In particular, if b = 1, the model is known as induced Ginibre ensemble, cf.…”