arXiv subjects

Mariya Shcherbina

Publications and source records attributed to Mariya Shcherbina.

15 recordsLinked to original sources

arXiv2026

Characteristic polynomials of non-Hermitian random band matrices near the threshold

The paper arXiv:2510.04255 shows that the asymptotic behavior of the second correlation function of characteristic polynomials of the $N\times N$ non-Hermitian random band matrices with a bandwidth $W$ exhibits the transition at $W\sim \sqrt{N}$, as $W,N\to \infty$: it coincides with those for Ginibre ensemble for $W\gg \sqrt{N}$, and factorized as $1\ll W\ll \sqrt{N}$. In this work we extend the techniques of arXiv:2510.04255 to study the critical regime when the bandwidth $W$ is proportional to the threshold $\sqrt{N}$.

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arXiv2025

Characteristic polynomials of non-Hermitian random band matrices

We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of a certain class of Gaussian $N\times N$ non-Hermitian random band matrices with a bandwidth $W$. Given $W,N\to\infty$, we show that this behavior near the point in the bulk of the spectrum exhibits the crossover at $W\sim \sqrt{N}$: it coincides with those for Ginibre ensemble for $W\gg \sqrt{N}$, and factorized as $1\ll W\ll \sqrt{N}$. The result is the first step toward the proof of Anderson's type transition for non-Hermitian random band matrices.

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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.

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arXiv2022

The least singular value of the general deformed Ginibre ensemble

We study the least singular value of the $n\times n$ matrix $H-z$ with $H=A_0+H_0$, where $H_0$ is drawn from the complex Ginibre ensemble of matrices with iid Gaussian entries, and $A_0$ is some general $n\times n$ matrix with complex entries (it can be random and in this case it is independent of $H_0$). Assuming some rather general assumptions on $A_0$, we prove an optimal tail estimate on the least singular value in the regime where $z$ is around the spectral edge of $H$ thus generalize the recent result of Cipolloni, Erdős, Schröder arxiv:1908.01653 to the case $A_0\ne 0$. The result improves the classical bound by Sankar, Spielman and Teng.

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arXiv2021

Finite-rank complex deformations of random band matrices: sigma-model approximation

We study the distribution of complex eigenvalues $z_1,\ldots, z_N$ of random Hermitian $N\times N$ block band matrices with a complex deformation of a finite rank. Assuming that the width of the band $W$ grows faster than $\sqrt{N}$, we proved that the limiting density of $\Im z_1,\ldots, \Im z_N$ in a sigma-model approximation coincides with that for the Gaussian Unitary Ensemble. The method follows the techniques of arXiv:1802.03813

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arXiv2019

Universality for 1 d random band matrices

We consider 1d random Hermitian $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \inΛ=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry's variance $J_{jk}=W^{-1}(δ_{j,k}+βΔ_{j,k})$ in each block. Considering the limit $W, n\to\infty$, we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as $W\gg \sqrt{N}$, is determined by the Wigner -- Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in [Shcherbina, M., Shcherbina, T.:Universality for 1d random band matrices: sigma-model approximation, J.Stat.Phys. 172, p. 627 -- 664 (2018)]

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arXiv2019

Transfer operator approach to 1d random band matrices

We discuss an application of the transfer operator approach to the analysis of the different spectral characteristics of 1d random band matrices (correlation functions of characteristic polynomials, density of states, spectral correlation functions). We show that when the bandwidth $W$ crosses the threshold $W=N^{1/2}$, the model has a kind of phase transition (crossover), whose nature can be explained by the spectral properties of the transfer operator.

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arXiv2018

Universality for 1d random band matrices: sigma-model approximation

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in J Stat Phys 164:1233 -- 1260, 2016; Commun Math Phys 351:1009 -- 1044, 2017. We consider random Hermitian block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \inΛ=[1,n]^d\cap \mathbb{Z}^d$) with a fixed entry's variance $J_{jk}=δ_{j,k}W^{-1}+βΔ_{j,k}W^{-2}$, $β>0$ in each block. Taking the limit $W\to\infty$ with fixed $n$ and $β$, we derive the sigma-model approximation of the second correlation function similar to Efetov's one. Then, considering the limit $β, n\to\infty$, we prove that in the dimension $d=1$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $β\gg n$, is determined by the classical Wigner -- Dyson statistics.

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arXiv2016

Transfer matrix approach to 1d random band matrices: density of states

We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Assuming that $n\ge CW\log W\gg 1$, we prove that the averaged density of states coincides with the Wigner semicircle law up to the correction of order $W^{-1}$.

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arXiv2016

Characteristic polynomials for 1D random band matrices from the localization side

We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by $J=(-W^2\triangle+1)^{-1}$. Assuming that the band width $W\ll \sqrt{n}$, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as $W, n\to \infty$) is equal to one, and so it does not coincides with those for GUE. This complements the previous result of T. Shcherbina and proves the expected crossover for 1D Hermitian random band matrices at $W\sim \sqrt{n}$ on the level of characteristic polynomials.

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arXiv2015

Fluctuations of the eigenvalue number in the fixed interval for $β$-models with $β=1,2,4$

We study the fluctuation of the eigenvalue number of any fixed interval $Δ=[a,b]$ inside the spectrum for $β$- ensembles of random matrices in the case $β=1,2,4$. We assume that the potential $V$ is polynomial and consider the cases of any multi-cut support of the equilibrium measure. It is shown that fluctuations become gaussian in the limit $n\to\infty$, if they are normalized by $π^{-2}\log n$.

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arXiv2015

On fluctuations of eigenvalues of random band matrices

We consider the fluctuation of linear eigenvalue statistics of random band $n\times n$ matrices whose entries have the form $\mathcal{M}_{ij}=b^{-1/2}u^{1/2}(|i-j|)\tilde w_{ij}$ with i.i.d. $w_{ij}$ possessing the $(4+\varepsilon)$th moment, where the function $u$ has a finite support $[-C^*,C^*]$, so that $M$ has only $2C_*b+1$ nonzero diagonals. The parameter $b$ (called the bandwidth) is assumed to grow with $n$ in a way that $b/n\to 0$. Without any additional assumptions on the growth of $b$ we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers [8] and [11], where CLT was proven under the assumption $n>>b>>n^{1/2}$. Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales, etc.

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arXiv2013

Change of variables as a method to study general $β$-models: bulk universality

We consider $β$ matrix models with real analytic potentials. Assuming that the corresponding equilibrium density $ρ$ has a one-interval support (without loss of generality $σ=[-2,2]$), we study the transformation of the correlation functions after the change of variables $λ_i\toζ(λ_i)$ with $ζ(λ)$ chosen from the equation $ζ'(λ)ρ(ζ(λ))=ρ_{sc}(λ)$, where $ρ_{sc}(λ)$ is the standard semicircle density. This gives us the "deformed" $β$-model which has an additional "interaction" term. Standard transformation with the Gaussian integral allows us to show that the "deformed" $β$-model may be reduced to the standard Gaussian $β$-model with a small perturbation $n^{-1}h(λ)$. This reduces most of the problems of local and global regimes for $β$-models to the corresponding problems for the Gaussian $β$-model with a small perturbation. In the present paper we prove the bulk universality of local eigenvalue statistics for both one-cut and multi-cut cases.

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arXiv2012

Fluctuations of linear eigenvalue statistics of $β$ matrix models in the multi-cut regime

We study the asymptotic expansion in $n$ for the partition function of $β$ matrix models with real analytic potentials in the multi-cut regime up to the $O(n^{-1})$ terms. As a result, we find the limit of the generating functional of linear eigenvalue statistics and the expressions for the expectation and the variance of linear eigenvalue statistics, which in the general case contain the quasi periodic in $n$ terms.

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arXiv2011

Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices

We consider two classical ensembles of the random matrix theory: the Wigner matrices and sample covariance matrices, and prove Central Limit Theorem for linear eigenvalue statistics under rather weak (comparing with results known before) conditions on the number of derivatives of the test functions and also on the number of the entries moments. Moreover, we develop a universal method which allows one to obtain automatically the bounds for the variance of differentiable test functions, if there is a bound for the variance of the trace of the resolvent of random matrix. The method is applicable not only to the Wigner and sample covariance matrices, but to any ensemble of random matrices.

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