On the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices

Abstract

We consider asymptotic behavior of the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices H n = n −1 A * m,n A m,n , where A m,n is a m × n complex matrix with independent and identically distributed entries ℜa αj and ℑa αj . We show that for the correlation function of any even order the asymptotic behavior in the bulk and at the edge of the spectrum coincides with those for the Gaussian Unitary Ensemble up to a factor, depending only on the fourth moment of the common probability law of entries ℜa αj , ℑa αj , i.e. the higher moments do not contribute to the above limit.

On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices

“…In the case of general Hermitian Wigner matrices it was proven that constant C N depends only on the first four moments of the matrix elements distribution and does not depend on any higher moments (see [15] for the case k = 1 and [24] for any k). The same result was obtained for general Hermitian sample covariance matrices (see [17] for the case k = 1 and [25] for any k). This shows that the local regime of correlation functions of characteristic polynomials is universal up to the first four moments.…”

“…In Section 2 we obtain a convenient integral representation for F 2 , using the integration over the Grassmann variables. The method is a generalization of that of [4,5] and is an analog of the method of [24,25], where the Hermitian Wigner and general sample covariance matrices were considered. In Section 3 we give the sketch of the proof of Theorem 1.…”

“…Characteristic polynomials of random matrices have been actively studied in the last years (see e.g. [1,5,6,14,15,16,18,20,24,25,27,29]). The interest to this topic is stimulated by its connections to the number theory, quantum chaos, integrable systems, combinatorics, representation theory and others.…”

Characteristic Polynomials for 1D Random Band Matrices from the Localization Side

“…They are especially convenient for the SUSY approach and were successfully studied by this techniques for many ensembles (see [5], [6], [16], [17], etc.). Although F 2k (Λ) is not a local object, it is also expected to be universal in some sense.…”

On the Correlation Functions of the Characteristic Polynomials of the Sparse Hermitian Random Matrices

“…The result was generalized soon on the correlation functions of any even order by T. Shcherbina in [22] where it was proposed the method which allowed to apply SUSY technique (or the Grassmann integration technique) to study the correlation functions of characteristic polynomials of random matrices with non Gaussian entries. The proposed method appeared to be rather powerful and since that was successfully applied to study characteristic polynomials of sample covariance matrices (see [23]) and band matrices [24,25].…”