Capturing noncommutativity in nonhomogeneous random matrices

Abstract

Random matrices are ubiquitous across many fields — physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure — entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). This thesis presents novel findings concerning the spectrum of such random matrices, which we say are nonhomogeneous. In particular, we focus on random matrices $X = \sum_{i=1}^n X_i$ such that $X_i$ are independent, but not necessarily identically distributed. First, we consider $X$ with independent Gaussian entries and an arbitrary variance profile. In this setting we show that $\norm{X}$ exhibits superconcetration, i.e. fluctuations of $\norm{X}$ are of smaller scale than that predicted by classical concentration inequalities. Moreover, we derive upper tail estimates for $\norm{X}$, which can be viewed as an extension of Tracy-Widom asymptotics for classical ensembles. Next, we show that if we instead assume that $\max_i \norm{X_i}$ has finite second moment, then under some fairly general conditions the spectrum of $X$ lies close to that of a Gaussian random matrix with the same mean and covariance. Whilst the proofs behind these facts differ substantially, the key idea underlying both arguments is to take advantage of noncommutativity of the summands $X_i$, rather than to find a way to treat $X_i$ as scalars, as was frequently done in earlier works on matrix concentration inequalities. As a consequence, we improve upon many of the previously known results for arbitrary $X$, as well as obtain novel conclusions in specialized settings, such as random graphs.