On the Correlation Functions of the Characteristic Polynomials of Real Random Matrices with Independent Entries

Abstract

The paper is concerned with the correlation functions of the characteristic polynomials of real random matrices with independent entries. The asymptotic behavior of the correlation functions is established in the form of a certain integral over unitary self-dual matrices with respect to the invariant measure. The integral is computed in the case of the second order correlation function. From the obtained asymptotics it is clear that the correlation functions behave like that for the Real Ginibre Ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries.

Phase transition of eigenvalues in deformed Ginibre ensembles

“…In this case it has the form X = X (1) X (2) −X (2) X (1) , (1.4) where both X (1) and X (2) are N × N complex matrices; see e.g. [41,55].…”

“…Some special cases of Corollary 2.5 have been studied by Afanasiev [1], Akemann-Phillips-Sommers [3], Grela [31], Forrester-Rains [25] and so on. In the case of the real Ginibre ensemble, that is, β = 1, Σ = Γ = I N , X 0 = 0, see [3] for the product of two characteristic polynomials and for the product of arbitrarily finite characteristic polynomials [1]. In the case of β = 2, Σ = Γ = I N and K 1 = K 2 , or general Σ, Γ and [31].…”

“…Next, we will make use of column replacement operations to calculate the above determinant det H (0) + H (1)…”

“…For this, we can take H (0) as a leading matrix and replace zero columns with corresponding non-zero columns from H (1) or H (2) by following the following rules:…”

Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity

“…entries. The correlations of characteristic polynomials in the latter two types of RMT ensembles have been under intensive studies in the last two decades, see [34][35][36][37][38][39][40][41][42][43][44][45].…”