Reblog: Random matrix theory and the Coulomb gas


inordinatum’s guest blog post here discusses something which, I must confess, isn’t going to be accessible to most of my readers. But it’s interesting to me, since it addresses many topics that are either directly in or close to my mathematical research interests.

The random matrix theory discussed here is the study of what we can say about matrices when we aren’t told the numbers in the matrix, but are told the distribution of the numbers — how likely any cell within the matrix is to be within any particular range. From that start it might sound like almost nothing could be said; after all, couldn’t anything go? But in exactly the same way that we’re able to speak very precisely about random events in the context of probability and statistics — for example, that a (fair) coin flipped a million times will come up tails pretty near 500,000 times, and will not come up tails 600,000 times — we’re able to speak confidently about the properties of these random matrices.

In any event, please do not worry about understanding the whole post. I found it fascinating and that’s why I’ve reblogged it here.

inordinatum

Today I have the pleasure of presenting you a guest post by Ricardo, a good physicist friend of mine in Paris, who is working on random matrix theory. Enjoy!

After writing a nice piece of hardcore physics to my science blog (in Portuguese, I am sorry), Alex asked me to come by and repay the favor. I am happy to write a few lines about the basis of my research in random matrices, and one of the first nice surprises we have while learning the subject.

In this post, I intent to present you some few neat tricks I learned while tinkering with Random Matrix Theory (RMT). It is a pretty vast subject, whose ramifications extend to nuclear physics, information theory, particle physics and, surely, mathematics as a whole. One of the main questions on this subject is: given a matrix $latex M$ whose entries are taken randomly from a…

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