arXiv subjects
Publications and source records attributed to Horng-Tzer Yau.
We consider the uniform random $d$-regular graph on $N$ vertices, with $d \in [N^α, N^{2/3-α}]$ for arbitrary $α> 0$. We prove that in the bulk of the spectrum the local eigenvalue correlation functions and the distribution of the gaps between consecutive eigenvalues coincide with those of the Gaussian Orthogonal Ensemble.
We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in the limit of large $N$.
We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $W\sim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.
We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times $t=o(1)$ with deterministic initial data V . Our main result states that if the density of states of $V$ is bounded both above and away from $0$ down to scales $\ell \ll t$ in a small interval of size $G \gg t$ around an energy $E_0$, then the local statistics coincide with the GOE/GUE near the energy $E_0$ after time $t$. Our methods are partly based on the idea of coupling two Dyson Brownian motions from [6], the parabolic regularity result of [15], and the eigenvalue rigidity results of [21].
We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.
We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v, (X^* X - z)^{-1} w \rangle - \langle v,w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w \in \mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z \geq N^{-1+ε}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
We consider the adjacency matrix of the ensemble of Erdős-Rényi random graphs which consists of graphs on $N$ vertices in which each edge occurs independently with probability $p$. We prove that in the regime $pN \gg 1$ these matrices exhibit bulk universality in the sense that both the averaged $n$-point correlation functions and distribution of a single eigenvalue gap coincide with those of the GOE. Our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances.
We introduce a class of $M \times M$ sample covariance matrices $\mathcal Q$ which subsumes and generalizes several previous models. The associated population covariance matrix $Σ= \mathbb E \cal Q$ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $Σ- I_M$ may depend on $M$ in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of $\mathcal Q$. We derive precise large deviation estimates on the generalized components $\langle \mathbf w, \boldsymbol ξ_i \rangle$ of the outlier and non-outlier eigenvectors $\boldsymbol ξ_i$. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the \emph{subcritical} spikes of $Σ$. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix $\mathcal Q$, satisfying $\mathbb E \mathcal Q = I_M$, as input: the isotropic local Marchenko-Pastur law established in [9], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution.
We consider generalized Wigner ensembles and general beta-ensembles with analytic potentials for any beta larger than 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian beta-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal.
We prove the edge universality of the beta ensembles for any $β\ge 1$, provided that the limiting spectrum is supported on a single interval, and the external potential is $\mathscr{C}^4$ and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class $\mathscr{C}^4$.
The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point $z$ away from the unit circle. More precisely, if $ | |z| - 1 | \ge τ$ for arbitrarily small $τ> 0$, the circular law is valid around $z$ up to scale $N^{-1/2+ \e}$ for any $\e > 0$ under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.
In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$ with $||z| - 1| > c $ for any $c>0$ independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $ |z|-1=\oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $ |z|-1=\oo(1)$ up to scale $N^{-1/4+ \e}$.
We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erdős-Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the $\ell^{\infty}$-norms of the $\ell^2$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $pN\gg N^{2/3}$.
We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in (\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \E |h_{xy}|^2$. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries $\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute $\E \abs{G_{xy}}^2$. We show that, as $L\to\infty$ and $W\gg L^{4/5}$, the behaviour of $\E |G_{xy}|^2$ is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.
We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [3,4,16].
We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of [5,16,17] which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices, and improve previous estimates from order 2 to order 4 in the cases relevant for applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper [3].
We prove the bulk universality of the $β$-ensembles with non-convex regular analytic potentials for any $β>0$. This removes the convexity assumption appeared in our earlier work. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a "convexified measure" so that the local statistics are preserved under this convexification.
We consider the ensemble of adjacency matrices of Erd{\H o}s-Rényi random graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption $p N \gg N^{2/3}$, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd{\H o}s-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd{\H o}s-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least $4 + ε$ moments.