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Yau, Horng-Tzer

Consider (N) bosons in a finite box (\Lambda= [0,L]^3\subset \mathbf R^3) interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle [\overline\lim_{\rho\to0} \overline\lim_{L \to \infty, N/L^3 \to \rho} \left(\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi a)^{5/2}(\rho)^{3/2}}\right)\leq \frac{16}{15\pi^2}, ] where (a) is the scattering length of the potential. Previously, an upper bound of the form (C 16/15\pi^2) for some constant (C > 1) was obtained in. Our result proves the upper bound of the the prediction by Lee-Yang and Lee-Huang-Yang.

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues {(x_{j})}({j=1}^{N}) are close to their classical location {(\gamma)({j})}({j=1}^{N}) determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/(N), the necessary apriori estimate on the location of eigenvalues requires only to know that (\mathbb{E}) |(x{j}) (-) (\gamma)(_{j})|(^{2}) (\leq) (N)(^{-1-\epsilon}) on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for Wishart matrices.

We consider N×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 introduce a class of M×MM×M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ=[E] Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IMΣ−IM may depend on MM in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of Q . We derive precise large deviation estimates on the generalized components [] of the outlier and non-outlier eigenvectors [] . 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 subcritical spikes of ΣΣ . The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q , satisfying [E] =IMEQ=IM , as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), 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 prove the universality of the β-ensembles with convex analytic potentials and for any β>0; that is, we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.

Consider (N × N) Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure (\nu_{ij}) with a subexponential decay. Let (\sigma_{ij}^2) be the variance for the probability measure (\nu_{ij}) with the normalization property that (\sum_i\sigma_{ij}^2 = 1) for all j. Under essentially the only condition that (c\leq N\sigma_{ij}^2 \leq c^{−1}) for some constant (c > 0), we prove that, in the limit (N \rightarrow \infty), the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale (M^{−1}).

Consider (N\times N) hermitian or symmetric random matrices (H) with independent entries, where the distribution of the ((i,j)) matrix element is given by the probability measure (\nu_{ij}) with zero expectation and with variance (\sigma_{ij}^2). We assume that the variances satisfy the normalization condition (\sum_{i} \sigma^2_{ij} = 1) for all (j) and that there is a positive constant (c) such that (c\le N \sigma_{ij}^2 \le c^{-1}). We further assume that the probability distributions (\nu_{ij}) have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of (H) is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order ( (N \eta)^{-1}) where (\eta) is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If (\gamma_j =\gamma_{j,N}) denotes the classical location of the (j)-th eigenvalue under the semicircle law ordered in increasing order, then the (j)-th eigenvalue (\lambda_j) is close to (\gamma_j) in the sense that for some positive constants (C, c) (\mathbb P \Big (\exists , j : ; |\lambda_j-\gamma_j| \ge (\log N)^{C\ log\ log\ N} \Big [ \min \big (, j, N-j+1 , \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{c\ log\ log\ N} \big]}) for (N) large enough. (2) The proof of the Dyson's conjecture which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order (N^{-1}). (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large (N) limit provided that the second moments of the two ensembles are identical.

We consider sample covariance matrices of the form X ∗X, where X is an M × N matrix with independent random entries. We prove the isotropic local MarchenkoPastur 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 hv,(X ∗X − z) −1wi − hv, wim(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w ∈ C N . We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z > N −1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

We consider the Ginzburg--Landau process on the lattice ℤdZd whose potential is a bounded perturbation of the Gaussian potential. We prove that the decay rate to equilibrium in the variance sense is t−d/2t−d/2 up to a~logarithmic correction, for any function uu with finite triple norm; that is, |||u|||=∑x∈ℤd‖∂ηxu‖∞<∞|||u|||=∑x∈Zd‖∂ηxu‖∞<∞.