Density of Small Singular Values of the Shifted Real Ginibre Ensemble

• Published in: Annales Henri Poincaré Vol. 23; no. 11; pp. 3981 - 4002
• Main Authors: Cipolloni, Giorgio, Erdős, László, Schröder, Dominik
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-11-2022; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Density; Dynamical Systems and Ergodic Theory; Eigenvalues; Elementary Particles; Mathematical and Computational Physics; Mathematical Methods in Physics; Original Paper; Parameters; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Theoretical; 15B52; 60B20; 68W40
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-022-01188-8

We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter
z
as the dimension tends to infinity. For
z
away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter
z
becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold.