Universal scaling limits of the symplectic elliptic Ginibre ensemble

Abstract

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behaviour of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel-Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.

Wronskian structures of planar symplectic ensembles

“…From the general universality principle of random matrix theory, it can be expected that for any fixed τ ∈ [0, 1), the local statistics of the ensemble in the large system coincide with the one (1.5) obtained from the case τ = 0. Based on the skew-orthogonal Hermite polynomials introduced by Kanzieper [36], such a statement was recently proved in [7] for p = 0 and in [20] for general p ∈ R.…”

“…Instead, we exploit the generalised Chirstoffel-Darboux formula introduced in [20], which allows us to obtain unified proofs for any points on the real line and to perform precise asymptotic analysis. (Indeed, this method can also be used to derive the subleading correction terms as well, see [20].)…”

“…Instead, we exploit the generalised Chirstoffel-Darboux formula introduced in [20], which allows us to obtain unified proofs for any points on the real line and to perform precise asymptotic analysis. (Indeed, this method can also be used to derive the subleading correction terms as well, see [20].) Furthermore, we describe the scaling limits in terms of the unified Wronskian form (1.4) and show that the associated kernels K of the form (1.8) correspond to those obtained from their complex counterparts.…”

“…Section 4 is devoted to the proofs of Theorems 2.1 and 2.2. In Subsection 4.1, we recall the Christoffel-Darboux formula in [20] and provide the general strategy of the proofs.…”

Truncations of random symplectic unitary matrices

“…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]. At the origin it also works for some non-Gaussian distributions, see [3] and references therein.…”