The epsilon-net method does indeed have some effectiveness when dealing with rectangular matrices (in which the spectrum stays well away from zero), but the situation becomes more delicate for square matrices it can control some “low entropy” portions of the infimum that arise from “structured” or “compressible” choices of, but are not able to control the “generic” or “incompressible” choices of, for which new arguments will be needed. Which sits at the “hard edge” of the spectrum, bears a superficial similarity to the operator normĪt the “soft edge” of the spectrum, that was discussed back in Notes 3, so one may at first think that the methods that were effective in controlling the latter, namely the epsilon-net argument and the moment method, would also work to control the former. As we shall see in the next set of notes, the least singular value of (and more generally, of the shifts for complex ) will be of importance in rigorously establishing the circular law for iid random matrices, as it plays a key role in computing the Stieltjes transform of such matrices, which as we have already seen is a powerful tool in understanding the spectra of random matrices. This quantity is also related to the condition number of, which is of importance in numerical linear algebra. Indeed, is invertible precisely when is non-zero, and the operator norm of is given by. This quantity controls the invertibility of. Now we turn attention to another important spectral statistic, the least singular value of an matrix (or, more generally, the least non-trivial singular value of a matrix with ). It is a key challenge to see how to weaken this joint independence assumption. These additional conditions were then slowly removed in a sequence of papers by Gotze-Tikhimirov, Girko, Pan-Zhou, and Tao-Vu, with the last moment condition being removed in a paper of myself, Van Vu, and Manjunath Krishnapur.Īt present, the known methods used to establish the circular law for general ensembles rely very heavily on the joint independence of all the entries. A rigorous proof of the circular law was then established by Bai, assuming additional moment and boundedness conditions on the individual entries. In 1984, Girko laid out a general strategy for establishing the result for non-gaussian matrices, which formed the base of all future work on the subject however, a key ingredient in the argument, namely a bound on the least singular value of shifts, was not fully justified at the time. In the case of random gaussian matrices, this result was established by Mehta (in the complex case) and by Edelman (in the real case), as was sketched out in Notes. This theorem has a long history it is analogous to the semi-circular law, but the non-Hermitian nature of the matrices makes the spectrum so unstable that key techniques that are used in the semi-circular case, such as truncation and the moment method, no longer work significant new ideas are required. Then the spectral measure converges both in probability and almost surely to the circular law, where are the real and imaginary coordinates of the complex plane. Theorem 1 (Circular law) Let be an iid matrix, whose entries, are iid with a fixed (complex) distribution of mean zero and variance one. For readers interested in this topic, I can recommend the recent Bourbaki exposé of Alice Guionnet.) (I had also hoped to discuss recent progress in eigenvalue spacing distributions of Wigner matrices, but have run out of time. In this final set of lecture notes for this course, we leave the realm of self-adjoint matrix ensembles, such as Wigner random matrices, and consider instead the simplest examples of non-self-adjoint ensembles, namely the iid matrix ensembles.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |