Least singular value, circular law, and Lindeberg exchange

These lectures cover three loosely related topics in random matrix theory. First we discuss the techniques used to bound the least singular value of (nonHermitian) random matrices, focusing particularly on the matrices with jointly independent entries. We then use these bounds to obtain the circular law for the spectrum of matrices with iid entries of finite variance. Finally, we discuss the Lindeberg exchange method which allows one to demonstrate universality of many spectral statistics of matrices (both Hermitian and non-Hermitian). 1. The least singular value This section1 of the lecture notes is concerned with the behaviour of the least singular value σn(M) of an n × n matrix M (or, more generally, the least nontrivial singular value σp(M) of a n×p matrix with p 6 n). This quantity controls the invertibility of M. Indeed, M is invertible precisely when σn(M) is non-zero, and the operator norm ‖M‖op of M−1 is given by 1/σn(M). This quantity is also related to the condition number σ1(M)/σn(M) = ‖M‖op‖M‖op of M, which is of importance in numerical linear algebra. As we shall see in Section 2, the least singular value of M (and more generally, of the shifts 1 √ n M− zI for complex z) will be of importance in rigorously establishing the circular law for iid random matrices M. The least singular value σn(M) = inf ‖x‖=1 ‖Mx‖, which sits at the “hard edge” of the spectrum, bears a superficial similarity to the operator norm ‖M‖op = σ1(M) = sup ‖x‖=1 ‖Mx‖ at the “soft edge” of the spectrum. For strongly rectangular matrices, the techniques that are useful to control the latter can also control the former, but the situation becomes more delicate for square matrices. For instance, the “epsilon net” 2010 Mathematics Subject Classification. Primary 60B20; Secondary 60F17.