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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., Comment: 33 pages

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 $|z_0|^2$., Comment: 31 pages, 1 figure

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, Erd\H{o}s, Schr\"{o}der arxiv:1908.01653 to the case $A_0\ne 0$. The result improves the classical bound by Sankar, Spielman and Teng., Comment: 25 pp

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, Comment: 30 pp. arXiv admin note: substantial text overlap with arXiv:1802.03813

This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the 1d band matrices to the case of the orthogonal symmetry. We consider $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \in\Lambda=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry's variance $J_{jk}=W^{-1}(\delta_{j,k}+\beta\Delta_{j,k})$ in each block. Considering the limit $W, n\to\infty$, we prove that the behavior of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold $W\sim \sqrt{N}$., Comment: 32 p

We consider 1d random Hermitian $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \in\Lambda=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry's variance $J_{jk}=W^{-1}(\delta_{j,k}+\beta\Delta_{j,k})$ 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)], Comment: 47 p

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., Comment: 19p. arXiv admin note: substantial text overlap with arXiv:1802.03813

The paper continues previous works which study the behavior of second correlation function of characteristic polynomials of the special case of $n\times n$ one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Applying the transfer matrix approach, we study the case when the bandwidth $W$ is proportional to the threshold $\sqrt{n}$, Comment: 21p

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 $j,k \in\Lambda=[1,n]^d\cap \mathbb{Z}^d$) with a fixed entry's variance $J_{jk}=\delta_{j,k}W^{-1}+\beta\Delta_{j,k}W^{-2}$, $\beta>0$ in each block. Taking the limit $W\to\infty$ with fixed $n$ and $\beta$, we derive the sigma-model approximation of the second correlation function similar to Efetov's one. Then, considering the limit $\beta, n\to\infty$, we prove that in the dimension $d=1$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $\beta\gg n$, is determined by the classical Wigner -- Dyson statistics., Comment: 38 pp

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 $W^{-1}$., Comment: 27 p

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., Comment: 30 p

We prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian real symmetric band matrices coincides with those for the Gaussian Orthogonal Ensemble (GOE). Here we adapt the approach of T. Shcherbina'14, where the case of 1D Hermitian random band matrices was considered., Comment: 29 p., paper is a continuation of arXiv:1209.3385

We consider the block band matrices, i.e. the Hermitian matrices $H_N$, $N=|\Lambda|W$ with elements $H_{jk,\alpha\beta}$, where $j,k \in\Lambda=[1,m]^d\cap \mathbb{Z}^d$ (they parameterize the lattice sites) and $\alpha, \beta= 1,\ldots, W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha\beta}$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha_1\beta_1}H_{j_2k_2,\alpha_2\beta_2}\rangle=\delta_{j_1k_2}\delta_{j_2k_1} \delta_{\alpha_1\beta_2}\delta_{\beta_1\alpha_2} J_{j_1k_1},$ where $J=1/W+\alpha\Delta/W$, $\alpha\le 1/4d$. This matrices are the special case of Wegner's $W$-orbital models. Assuming that the number of sites $|\Lambda|$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda_0|< \sqrt{2}$., Comment: some typos were fixed

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices, i.e. of the hermitian matrices $H_n$ with independent Gaussian entries such that $< H_{ij}H_{lk}>=\delta_{ik}\delta_{jl}J_{ij}$, where $J=(-W^2\triangle+1)^{-1}$. Assuming that $W^2=n^{1+\theta}$, $0<\theta<1$, we show that this asymptotic behavior (as $n\to\infty$) in the bulk of the spectrum coincides with those for the Gaussian Unitary Ensemble., Comment: 35 pages

We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$., Comment: 25 pages, 2 figures

We consider the asymptotics of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$. We show that for the correlation function of any even order the asymptotic coincides with this for the GUE up to a factor, depending only on the forth moment of the common probability law $Q$ of entries $\Im W_{jk}$, $\Re W_{jk}$, i.e. that the higher moments of $Q$ do not contribute to the above limit., Comment: 20 p