MathSciDoc: An Archive for Mathematician ∫

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|≥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 this paper we prove the universality of covariance matrices of the form $H_{Ntimes N}={X}^{dagger}X$ where $X$ is an ${Mtimes N}$ rectangular matrix with independent real valued entries $x_{ij}$ satisfying $mathbb{E}x_{ij}=0$ and $mathbb{E}x^2_{ij}={frac{1}{M}}$, $N$, $Mto infty$. Furthermore it is assumed that these entries have sub-exponential tails or sufficiently high number of moments. We will study the asymptotics in the regime $N/M=d_Nin(0,infty),lim_{Ntoinfty}d_Nneq0,infty$. Our main result is the edge universality of the sample covariance matrix at both edges of the spectrum. In the case $lim_{Ntoinfty}d_N=1$, we only focus on the largest eigenvalue. Our proof is based on a novel version of the Green function comparison theorem for data matrices with dependent entries. En route to proving edge universality, we establish that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko-Pastur law uniformly up to the edges of the spectrum with an error of order $(Neta)^{-1}$ where $eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. Combining these results with existing techniques we also show bulk universality of covariance matrices. All our results hold for both real and complex valued entries.

We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we identify the law of the outliers and prove that the non-outliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the \emph{isotropic local semicircle law}, which establishes optimal high-probability bounds on the quantity $< v,[(H - z)^{-1} - m(z) 1] w >$, where $m(z)$ is the Stieltjes transform of Wigner's semicircle law and $v, w$ are arbitrary deterministic vectors.

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 $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

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 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 [17,19,6].

We derive an upper bound on the free energy of a Bose gas at density $ \varrho $ and temperature T. In combination with the lower bound derived previously by Seiringer (Commun. Math. Phys. 279(3): 595–636, 2008), our result proves that in the low density limit, i.e., when $ a^3 \varrho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$, the free energy difference per volume between interacting and ideal Bose gases, is equal to $ 4\pi a(2\varrho^{2}-[\varrho-\varrho_{c}]^{2}_{+}) $. Here, $\varrho_c (T)$ denotes the critical density for Bose–Einstein condensation (for the ideal Bose gas), and $ [⋅]_{+}= max \{⋅ , 0\}$ denotes the positive part.

The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is $ 2\pi \hbar^2 a \varrho/m$ when the density of the gas is $ \varrho$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a > 0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.

Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box  of side length $L$. We take the $limit N,L \to \infty $ while keeping the density $\rho = N/L^3 fixed and small. We prove a new lower bound for its ground state energy per particle $$ \frac{E(N,\Lambda)}{N} \ge 4 \pi a \rho [1- O(\rho^{1/3} | log \rho |^3) ] $$ as $\rho \to 0$, where $a$ is the scattering length of $V$.