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 $G(n,\frac{p}{n})$ Erd{\H o}s -- Rényi 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 $(-{\lambda }_{\ast }(p),{\lambda }_{\ast }(p))$ such that for ${\lambda }_0\in (-{\lambda }_{\ast }(p),{\lambda }_{\ast }(p))$ 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.

32 pages; moved subsection on Grassmann variables to Appendix, replaced the proof of lemma 1 by more elegant one, added some references, corrected typos

References in corpus (4)

• Spectra of Sparse Random Matrices
• Bulk universality of sparse random matrices
• On absolute moments of characteristic polynomials of a certain class of complex random matrices
• On the Second-Order Correlation Function of the Characteristic Polynomial of a Hermitian Wigner Matrix

Cited by in corpus (4)

• Characteristic polynomials for 1D random band matrices from the localization side
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• Characteristic polynomials for random band matrices near the threshold
• Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity