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

Abstract

We consider asymptotics of the correlation functions of characteristic polynomials corresponding to random weighted G(n, p n ) Erdős-Rényi graphs with Gaussian weights in the case of finite p and also when p → ∞. It is shown that for finite p the second correlation function demonstrates a kind of transition: when p < 2 it factorizes in the limit n → ∞, while for p > 2 there appears an interval (−λ * ( p), λ * ( p)) such that for λ 0 ∈ (−λ * ( p), λ * ( p)) the second correlation function behaves like that for Gaussian unitary ensemble (GUE), while for λ 0 outside the interval the second correlation function is still factorized. For p → ∞ there is also a threshold in the behavior of the second correlation function near λ 0 = ±2: for p n 2/3 the second correlation function factorizes, whereas for p n 2/3 it behaves like that for GUE. For any rate of p → ∞ the asymptotics of correlation functions of any even order for λ 0 ∈ (−2, 2) coincide with that for GUE.

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

“…Moreover, the correlation functions of the characteristic polynomials are of independent interest. They were studied for many ensembles of Hermitian and real symmetric matrices, for instance [1,11,12,51,53,54] etc. The other result on the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian matrices of the form H + iΓ, where H is from Gaussian Unitary Ensemble (GUE) and Γ is a fixed matrix of rank M , was obtained in [28].…”

Untitled

“…Moreover, the correlation functions of the characteristic polynomials are of independent interest. They were studied for many ensembles of Hermitian and real symmetric matrices, for instance, [1,8,9,34,36,37] etc. Let us introduce the m th correlation function of the characteristic polynomials…”

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].…”