Talks

I've given several expository talks at Newcastle's postgraduate forum, on a variety of topics:

Notelets

The following are some snippets of known material that I've written up, either for my own understanding or because I couldn't find an explicit reference.

And now for something completely different

Several years ago, as part of my fourth-year studies at Cambridge University, I wrote what was supposed to be a survey essay on ``Random Matrices". The end result wasn't much of a survey in the end, but did present a proof of almost-sure convergence of the spectral measure for the GOE. Below are copies of the essay, LaTeXed from the original source file.

Part III Essay: Random matrices and convergence to the semicircle law (2001), 18 pp.
Cover page (DVI) | Essay (DVI)
Cover page (gzipped PS) | Essay (gzipped PS)

Feel free to make use of the contents (caveat lector!), but if you do please acknowledge my work and the work I've cited.

Some remarks on the essay

Since writing this essay I've become aware of some gaps in my reading of the literature, so if you have any feedback on possible improvements and correct attribution please let me know. Any future revision of this essay will take such comments into account.

The most important oversight is that the idea of tridiagonalising the GOE was explored in a 1984 paper of Trotter:

Trotter, Hale F. Eigenvalue distributions of large Hermitian matrices; Wigner's semicircle law and a theorem of Kac, Murdock, and Szegö.
Adv. in Math. 54 (1984), no. 1, 67--82. [MR 86c:60055]

What I wrote isn't complete as an essay -- there was a chapter and coda that I had planned to add, but which I never hammered out in time for the submission deadline. However, the proof I give of the main result may still be of interest: it combines ideas of Pastur (working with Cauchy transforms rather than explicit moment estimates) and Haagerup and Thorbjørnsen (using a Gaussian concentration inequality).

Neither of these ideas is new, but it seems to have been overlooked or little mentioned that one can play the two off against each other. To paraphrase: rather than first calculate the limit of the expected spectral measure and then use concentration to get a.s. convergence, the proof in the essay first establishes concentration and then uses that to get a limiting functional equation for (the Cauchy transform of) the expected spectral measures. (The idea of getting a limiting functional equation is used in Pastur's papers, but where we use concentration he uses a direct and rather messy estimate on the variance.)

(Added 29-08-05) Proper revision of the essay, including an all-important update of the background and context, is sadly not something I will get round to in the near future. I should at least remark that the theme of proving concentration around the mean empirical distribution, rather than identifying the limit and proving a.s. convergence to it by hand, seems to have been taken up and clarified nicely by several authors. In particular, recent work of Sourav Chatterjee appears to warrant close reading.

As a final remark: if you are reading the essay, I hope that the deliberately informal style of writing does not grate, frustrate, or infuriate.