Category Archives: Random Matrices 2019/2020

Correction on my lecture notes on random matrices (weak convergence versus convergence of moments)

I just noticed that I have a stupid mistake in my random matrix lecture notes (and also in the recording of the corresponding lecture). I am replacing the notes with a new version which corrects this.

In Theorem 4.16, I was claiming that weak convergence is equivalent to convergence of moments, in a setting where all moments exist and the limit is determined by its moments. Of course, this is a too optimistic statement. What is true is the direction that convergence of moments implies weak convergence. That’s the important direction. The other direction would be more of a relevance for combinatorial aficionados like me, as it would allow me to claim that the combinatorial and the analytical approach in such a setting are equivalent. However, the other direction is clearly wrong without some additional assumptions; and thus there are nice-looking situations where one cannot prove weak convergence by dealing with moments.

Of course this is not a new insight. In the context of proving the convergence to the semicircle for Wigner matrices with non-zero mean for the entries we know that we cannot do this with moments (see for example, Remark 11 in Chapter 4 of my book with Jamie).

To get a kind of positive spin out of this annoying mistake, I started to think about what kind of convergence we actually want in our theorems in free probability. Usually our convergence is in distribution, i.e., we are looking on moments – which seems to be the natural thing to do in the multivariate case of several non-commuting operators. However, we can also project things down to the classical world of one variable by taking functions in our operators and ask for the convergence of all such functions. And then there might be a difference whether we ask for weak convergence or for convergence in distribution (i.e., convergence of all moments).

This might become kind of relevant in the context of rational functions. Sheng Yin showed in Non-commutative rational functions in strong convergent random variables that convergence in distribution goes over from polynomials to rational functions (in the case where we assume that the rational function in the limit is a bounded operator) if we assume strong convergence on the level of polynomials (i.e., also convergence of the operator norms). Without the assumption of strong convergence it is easy to see that there are examples (see page 12 of the paper of Sheng) where one has convergence in distribution for the polynomials, but not for the rational functions. However, though one does not have convergence of the moments of the rational function, it is still true in this example that one has weak convergence of the (selfadjoint) rational function. So maybe it could still be the case that, even without strong convergence assumptions, convergence in distribution for polynomials (or maybe weak convergence for polynomials) implies weak convergence for rational functions. At least at the moment we do not know a counter example to this.

The saga ends …

I have now finished my class on random matrices. The last lecture motivated the notion of (asymptotic) freeness from the point of view of looking on independent GUE random matrices. So you might think that there should now be continuations on free probability and alike coming soon. But actually this part of the story was already written and recorded and if you don’t want to spoil the tension you should watch the series not in its historical but in its logical order:

  1. Random Matrices (videos, homepage of class)
  2. Free Probability Theory (videos, homepage of class)
  3. Non-commutative Distributions (and Operator-Valued free Probability Theory) (videos, homepage of class)

More information, in particular the underlying script (sometimes in a handwritten version, sometimes in a more polished texed version), can be found on the corresponding home page of the lecture series.

May freeness be with you …

Class on “Random Matrices”, Winter Term 2019/20

Our winter term has just started, running from mid October 2019 to mid February 2020, with a two-week break around Christmas. This term I am giving an introduction to random matrices. Again, the lectures will be recorded and put online. The lectures can be found on our video platform; more info on the lectures are also on the website of the class.

The lectures will follow roughly the material from the same class of summer term 2018, for which there exist also texed lecture notes. There will be a few reorganizations and shifts in the material, so there might emerge also a new version of the lectures notes sometimes in the future …