Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit

For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $$\mathfrak C_N(T)= \mathbb E\bigg[ \prod_{(i,j) \in T} \Big(\mathbf 1_{\big(\{i,j\} \in G_N \big)} - \mathbb P(\{i,j\} \in G_N) \Big)\bigg], \ T \subset [N]^2 \textrm{finite}$$ furnishes informations on the asymptotic spectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N = N\times \mathbb P(\{i,j\} \in G_N)$ and assume $d_N, N-d_N\underset{N \rightarrow \infty}{\longrightarrow} \infty$. If $\mathfrak C_N(T) =\big(\frac{d_N}N\big)^{|T|} \times O\big(d_N^{-\frac {|T|}2}\big)$ for any $T$, the standardized empirical eigenvalue distribution of $A_N$ converges in expectation to the semicircular law and the matrix satisfies asymptotic freeness properties in the sense of free probability theory. We provide such estimates for uniform $d_N$-regular graphs $G_{N,d_N}$, under the additional assumption that $|\frac N 2 - d_N- \eta \sqrt{d_N}| \underset{N \rightarrow \infty}{\longrightarrow} \infty$ for some $\eta>0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.Comment: 21 pages, 7 figure

Limit processes for TASEP with shocks and rarefaction fans

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.Comment: 46 pages, 3 figures, LaTeX; Extended explanations in the first two section

Eigenvectors of some large sample covariance matrix ensembles

We consider sample covariance matrices $${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$$ where X N is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and ΣN is a N × N positive definite matrix. In addition we assume that the spectral measure of ΣN almost surely converges to some limiting probability distribution as N → ∞ and p/N → γ >0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type $${\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}$$ where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its invers

Convergence of the two-point function of the stationary TASEP

We consider the two-point function of the totally asymmetric simple exclusion process with stationary initial conditions. The two-point function can be expressed as the discrete Laplacian of the variance of the associated height function. The limit of the distribution function of the appropriately scaled height function was obtained previously by Ferrari and Spohn. In this paper we show that the convergence can be improved to the convergence of moments. This implies the convergence of the two-point function in a weak sense along the near-characteristic direction as time tends to infinity, thereby confirming the conjecture in the paper of Ferrari and Spohn.Comment: LaTeX, 18 pages, 2 figures; Minor correction

of band or sparse random matrices

In this text, we consider an random N × N matrix X such that all but o(N) rows of X have W non identically zero entries, the other rows having less than W entries (such as, for example, standard or cyclic band matrices). We always suppose that 1 ≪ W ≪ N. We first prove that if the entries are independent, centered, have variance one, satisfy a certain tail upper-bound condition and W ≫ (log N) 6(1+α), where α is a positive parameter depending on the distribution of the entries, then the largest eigenvalue of X / √ W converges to the upper bound of its limit spectral distribution, that is 2, as for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin where less hypotheses were made on W, but more hypotheses were made about the law of the entries and the structure of the matrix. Then, under the same hypotheses, we prove a delocalization result for the eigenvectors of X. More precisely we show that eigenvectors associated to eigenvalues “far enough” from zero cannot be essentially localized on less than W / log(N) entries. This lower bound on the localization length has to be compared to the recent result by Steinerberger, which states that the localization length in the edge is ≪ W 7/5 or there is strong interaction between two eigenvectors in an interval of length W 7/5

Corrigendum: Limit Process of Stationary TASEP near the Characteristic Line

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/96244/1/21439_ftp.pd

The largest eigenvalue of rank one deformation of large Wigner matrices

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration

The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case

We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered