On the correlation functions of the characteristic polynomials of the sparse hermitian random matrices

TL;DR

This paper analyzes the asymptotic behavior of correlation functions of characteristic polynomials in sparse Hermitian random matrices, revealing phase transitions depending on the sparsity parameter p and the matrix size n.

Contribution

It provides a detailed description of the transition in correlation function behavior for Erdős-Rényi graphs with Gaussian weights as p varies, including finite and infinite regimes.

Findings

For finite p, the second correlation function transitions from factorized to GUE-like behavior as p crosses 2.

As p approaches infinity, the correlation functions exhibit a phase transition near p ~ n^{2/3}.

Correlation functions of even order for λ₀ in (-2, 2) match GUE asymptotics regardless of p growth rate.

Abstract

We consider asymptotics of the correlation functions of characteristic polynomials corresponding to random weighted $G(n, \frac{p}{n})$ Erd{\H o}s -- R\'enyi graphs with Gaussian weights in the case of finite $p$ and also when $p \to \infty$. It is shown that for finite $p$ the second correlation function demonstrates a kind of transition: when $p < 2$ it factorizes in the limit $n \to \infty$, while for $p > 2$ there appears an interval $(-\lambda_*(p), \lambda_*(p))$ such that for $\lambda_0 \in (-\lambda_*(p), \lambda_*(p))$ the second correlation function behaves like that for GUE, while for $\lambda_0$ outside the interval the second correlation function is still factorized. For $p \to \infty$ there is also a threshold in the behavior of the second correlation function near $\lambda_0 = \pm 2$: for $p \ll n^{2/3}$ the second correlation function factorizes, whereas for $p \gg n^{2/3}$ it behaves like that for GUE. For any rate of $p \to \infty$ the asymptotics of correlation functions of any even order for $\lambda_0 \in (-2, 2)$ coincide with that for GUE.