arXiv subjects

Ievgenii Afanasiev

Publications and source records attributed to Ievgenii Afanasiev.

6 recordsLinked to original sources

arXiv2026

Asymptotic behavior of eigenvalues of large rank perturbations of large random matrices

The paper is concerned with deformed Wigner random matrices. These matrices are closely related to Deep Neural Networks (DNNs): weight matrices of trained DNNs could be represented in the form $R + S$, where $R$ is random and $S$ is highly correlated. The spectrum of such matrices plays a key role in rigorous underpinning of the novel pruning technique based on Random Matrix Theory. In practice, the spectrum of the matrix $S$ can be rather complicated. In this paper, we develop an asymptotic analysis for the case of full rank $S$ with increasing number of outlier eigenvalues.

math-ph↗

arXiv2025

Universality of the second correlation function of the deformed Ginibre ensemble

We study the deformed complex Ginibre ensemble $H=A_0+H_0$, where $H_0$ is the complex matrix with iid Gaussian entries, and $A_0$ is some general $n\times n$ matrix (it can be random and in this case it is independent of $H_0$). Assuming rather general assumptions on $A_0$, we prove that the asymptotic local behavior of the second correlation function of the eigenvalues of such matrices in the bulk coincides with that for the pure complex Ginibre ensemble.

math-ph↗

arXiv2023

Characteristic polynomials of sparse non-Hermitian random matrices

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 $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalised iid Bernoulli$(p)$ $d_{jk}$. It is shown that, as $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for Ginibre ensemble: it converges to a determinant with Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. For the finite $p>0$, the behavior is different and exhibits the transition between three different regimes depending on values of $p$ and $|z_0|^2$.

math-ph↗

arXiv2022

On the Correlation Functions of the Characteristic Polynomials of Random Matrices with Independent Entries: Interpolation Between Complex and Real Cases

The paper is concerned with the correlation functions of the characteristic polynomials of random matrices with independent complex entries. We investigate how the asymptotic behavior of the correlation functions depends on the second moment of the common probability law of the matrix entries, a sort of ``reality measure'' of the entries. It is shown that the correlation functions behave like that for the Complex Ginibre Ensemble up to a factor depending only on the second moment and the fourth absolute moment of the common probability law of the matrix entries.

math-ph↗

arXiv2019

On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries

The paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian random matrices with independent entries. It is shown that the correlation functions behave like that for the Complex Ginibre Ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries.

math-ph↗

arXiv2016

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

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 $(-λ_*(p), λ_*(p))$ such that for $λ_0 \in (-λ_*(p), λ_*(p))$ the second correlation function behaves like that for GUE, while for $λ_0$ 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 $λ_0 = \pm 2$: 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 $λ_0 \in (-2, 2)$ coincide with that for GUE.

math-ph↗