On the almost‐circular symplectic induced Ginibre ensemble

• Published in: Studies in applied mathematics (Cambridge) Vol. 150; no. 1; pp. 184 - 217
• Main Authors: Byun, Sung‐Soo, Charlier, Christophe
• Format: Journal Article
• Language: English
• Published: Cambridge Blackwell Publishing Ltd 01.01.2023
• Subjects: asymptotic analysis; Constraining; Correlation; Kernels; Matematik; Matematisk analys; Mathematical Analysis; Mathematics; Natural Sciences; Naturvetenskap; Probability Theory and Statistics; random matrix theory; Sannolikhetsteori och statistik; universality
• ISSN: 0022-2526; 1467-9590
• DOI: 10.1111/sapm.12537

We consider the symplectic‐induced Ginibre process, which is a Pfaffian point process on the plane. Let N be the number of points. We focus on the almost‐circular regime where most of the points lie in a thin annulus SN$\mathcal {S}_{N}$ of width O1N$O\left(\frac{1}{N}\right)$ as N→∞$N \rightarrow \infty$. Our main results are the bulk scaling limits of all correlation functions near the real axis, and also away from the real axis. Near the real axis, the limiting correlation functions are Pfaffians with a new correlation kernel, which interpolates the limiting kernels in the bulk of the symplectic Ginibre ensemble and of the antisymmetric Gaussian Hermitian ensemble of odd size. Away from the real axis, the limiting correlation functions are determinants, and the kernel is the same as the one appearing in the bulk limit of almost‐Hermitian random matrices. Furthermore, we obtain precise large N asymptotics for the probability that no points lie outside SN$\mathcal {S}_{N}$, as well as of several other “semi‐large” gap probabilities.