## Random Matrices and the Riemann Hypothesis

January 15, 2010

I made it to my second post!  This is the greatest achievement of my blogging life so far.

Having never even heard of random matrix theory before last week, I have recently been hearing talk of it bandied about all over the place.  When I asked my supervisor why this might be, he replied that they were a “hot topic”, and went on to explain the recent connections that have been made between the distribution of zeros of the Riemann zeta function, and that of eigenvalues of large random matrices.  So I’m going to write a bit about it here.

Random matrices are intuitively exactly what you might first suspect they are: matrices with “random” entries.  Of course, it is technically meaningless to say that an element of a matrix is random (hence the inverted commas), but what we can do is to use probability theory to rigorously define an analogous concept.  Without going into in too much depth, we can think of a random matrix as being a matrix with elements that are random numbers from some probability distribution (that is, random variables).  So an $n\times n$ random matrix is actually a collection of all the possible $n\times n$ matrices, along with probability density functions telling us how likely each of these is to occur.