On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries

Abstract

The paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian random matrices with independent entries. It is shown that the correlation functions behave like that for the Complex Ginibre Ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries.

Untitled

“…Random matrices with independent entries are usually considered over complex numbers, real numbers or quaternions. An asymptotic behavior of the correlation functions of the characteristic polynomials was recently computed in the complex case [2]. The goal of the current paper is to establish a similar result in the real case.…”

“…a (Ξ, Φ, Θ) is a polynomial and its every monomial has a degree at least 2 with respect to Ξ and at least 2 with respect to ϕ k and ϑ k . Put p (2) a (Ξ,…”

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

“…Here, sophisticated techniques as the Riemann-Hilbert method have been developed. The universality of products and ratios of characteristic polynomials has been directly addressed as well, yielding a generating functional for the kernel, both for invariant [16] and Hermitian Wigner ensembles [17], see also [26,1] for recent work using supersymmetry. We refer to [28,7] for most concise expressions for averages of products and ratios of characteristic polynomials at finite-N , and to [18] for the supersymmetric perspective on that.…”

“…In [3] it was shown for complex matrices with unit variances σ k = 1 that the squared singular values of the product matrix G 1 • • • G M form a determinantal point process, representing an example for a polynomial ensemble. The corresponding kernel of biorthogonal functions was explicitly determined in [3] using Gram-Schmidt orthogonalisation 1 . As the Heine formula trivially extends to polynomial ensembles, the following holds for the orthogonal polynomials:…”

Untitled

“…(ii) A direct argument from the stationary growth model to the Busemann limit was introduced in [Georgiou et al 2015] for the log-gamma polymer, and applied to the exponential CGM in the lecture notes [Seppäläinen 2018]. An application of this strategy to the CGM with general i.i.d.…”