Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

Abstract

The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly nonsymmetric.

GUE-chGUE transition preserving chirality at finite matrix size

“…We would like to point out that similar Pfaffian structures have been derived for several other two matrix models [48,49,50,51,52,53,54,55,44,56,57]. Considering the fact that determinantal point processes can also be rewritten as Pfaffian ones [65], Pfaffian point processes seem to be more natural than determinantal point processes.…”

Topology trivialization transition in random non-gradient autonomous ODEs on a sphere

“…The density function ρ (r) N (x) is known in the closed form in terms of Hermite polynomials [29]. Assuming for simplicity that N is even, one has ρ (r)…”

“…Based on this expression, all relevant asymptotics of ρ (r) N (λ √ N ) for large N 1 were worked out in [29] and [12]. This allows one to carry out an asymptotic evaluation of the integral in (2.14) in various regimes of parameters τ and b 2 in the limit N → ∞.…”

Dirac Spectra in Dense QCD

“…This 'rotated photon' is unscreened in this phase. Despite serious efforts so far, it is still under a debate (see [62][63][64] for recent works). This phase is termed as the "Color-Flavor-Locked (CFL) phase" [53].…”

“…The complete solution of all real and complex eigenvalue correlations was obtained only recently by several groups [60][61][62][63][64][65]. The case β = 1, the Gaussian ensemble of real asymmetric matrices, turned out to be the most difficult one because the joint eigenvalue distribution function is not absolutely continuous; for finite N , there is always nonzero probability of finding real eigenvalues in addition to complex eigenvalues occurring in complex conjugate pairs.…”