The dimension-free structure of nonhomogeneous random matrices

Abstract

Let X be a symmetric random matrix with independent but nonidentically distributed centered Gaussian entries. We show thatwhere Sp denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case p = ∞, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on ℓ 2 . Along the way, we obtain optimal dimension-free bounds on the moments (E X p Sp ) 1/p that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.2000 Mathematics Subject Classification. 60B20; 46B09; 46L53; 15B52.

Outliers in spectrum of sparse Wigner matrices

“…Whenever , it is known that with probability tending to one with n . This result was verified in a series of works where the assumptions on the matrix sparsity and the matrix entries were sequentially relaxed (see [7,22,28,32,49]). On the other hand, when p n → 0 with n relatively fast, one can easily verify that the extreme eigenvalues get asymptotically detached from the bulk of the spectrum.…”

“…More generally, when , this phenomenon was observed in the case of Rademacher variables [28] and in the case of the Erdős–Renyi random graphs [6] which will be discussed later on. In the window around it is known that, up to constant multiples, the matrix norm is of order [6,7,32,46], however, to the best of our knowledge, there have been no results on its exact asymptotic behavior. In this connection, we can ask the following questions: Is there a sharp phase transition (in terms of sparsity) in the appearance/disappearance of outliers in the semi‐circular law? For concrete distributions, say, sparse Bernoulli matrices, what is the explicit formula for the sparsity threshold (if it exists)? What is a conceptual explanation of why the outliers appear at a particular level of sparsity? What is the exact asymptotic value of an outlier? …”

Estimates of norms of log-concave random matrices with dependent entries

“…The next corollary is a version of Theorem 1.1 in the spirit of the aforementioned results from [17,14,16]. It follows directly from (1.3), and the Jensen inequality.…”

“…when X ij " a ij g ij and g ij are i.i.d. standard Gaussian variables), as was recently shown by Lata la, van Handel, and Youssef in [14], and up to a logarithmic factor for any X with independent centred entries, see [16]. The advance of the two latest results is that they do not require that the entries of X are equally distributed (nor that they have equal variances).…”

An upper bound on the smallest singular value of a square random matrix

“…This procedure relies on an upper bound for the operator norm B , which is of order n 1 2 with exponentially high probability under the subgaussian moment assumption on the entries of B. Moreover, as can be seen in [7,39], one has that B ≤ C √ n under the assumption of bounded fourth moments (see [10,11] for independent but not identically distributed entries). However, in the settings of Theorem 1.1, it is not guaranteed that the operator norm B has a good upper bound.…”