Characteristic polynomials of sparse non-Hermitian random matrices

2023/12/15 by Afanasiev, Ievgenii, Shcherbina, Tatyana · 2 citations
#15B52 #60B20 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2312.10220

Abstract

We consider the asymptotic local behavior of the second correlation function of the characteristic polynomials of sparse non-Hermitian random matrices X_n whose entries have the form x_jk=d_jkw_jk with iid complex standard Gaussian w_jk and normalised iid Bernoulli(p) d_jk. It is shown that, as p→∞, the local asymptotic behavior of the second correlation function of characteristic polynomials near z₀∈ ℂ coincides with those for Ginibre ensemble: it converges to a determinant with Ginibre kernel in the bulk |z₀|<1, and it is factorized if |z₀|>1. For the finite p>0, the behavior is different and exhibits the transition between three different regimes depending on values of p and |z₀|².

Cited by

• Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
• Characteristic polynomials of non-Hermitian random band matrices

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