From random matrix to beta gamma algebra

Today I started reading Terence Tao’s blog on universality of Wigner eigenvalue distribution up to the edge and he mentioned an interesting distribution, namely that of \lambda_i(M_n) \in [\lambda_i(M_{n+1}),\lambda_{i+1}(M_{n+1})], when the eigenvalues of M_{n+1} are held fixed but the eigenvectors are uniformized under the natural SU(n) action, and where \lambda_i is the ith largest eigenvalue and M_n is the bottom n \times n minor of the (n+1) \times (n+1) dimenional Hermitian matrix M_{n+1}. The implied inequalities follow from Cauchy’s interlacing formula, which is in turn a consequence of the minimax characterization of the ordered eigenvalues of a Hermitian matrix. He mentioned that for general n, this distribution has a piecewise polynomial density, which reminds me of the result of Duistermaat-Heckman formula, of which I knew little beyond the fact that there is also a piecewise polynomial floating around about the image of the moment map.

Then Terence specialized briefly to the case of n=1, of which he mentions the distribution above becomes uniform, and in fact has connections with some famous result of Archimedes relating the surface area of the 2-sphere and that of the cylinder. Here I will attempt to elaborate on this point.

First of all we can write the matrix M_2 as
\begin{pmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{pmatrix}. And to say the eigenvalues stay the same but the eigenvectors is uniform over SU(2) is the same as conjugating by a uiform SU(2) element, and then look at the bottom right element which corresponds to M_1. If we represent SU(2) by
\begin{pmatrix} \alpha & - \bar{\beta} \\ \beta & \bar{\alpha} \end{pmatrix}, then the bottom right corner becomes |\beta|^2 \lambda_1 + |\alpha|^2 \lambda_2, which leaves us the task of computing the distribution of |\alpha|^2, under the haar distribution on SU(2). The standard probabilistic way of doing this is through so-called beta gamma algebra.

Here are the ingredients: by normalizing a standard Gaussian random vector, one gets uniform measure on the unit sphere, i.e., if X_1, \ldots, X_n are iid standard normal, then Y_i := \frac{X_i}{ \sqrt{\sum_{j=1}^n X_j^2}} are the coordinates of a uniformly chosen point on S^{n-1}. The partial sums of squares of the coordinates are in fact beta random variables, which follows from the following two facts:

1. square of Gaussian gives an exponentially distributed random variable, which is also a gamma(1/2) variable. Sum of independent gammas become gamma with parameter the sum of the respective parameters.

2. if we have two gammas X_1, X_2 of parameter \gamma_1, \gamma_2 respectively, then \frac{X_1}{X_1 + X_2} is a beta variable, with parameter \gamma_1, \gamma_2, whose proof requires integration involving in general independent gamma variables, of which the key step is to do a clever n-diimensional change of variables, obtained by finding the determinant of an upper triangular matrix (without seeing this trick, the computation can seem rather horrendous)

Now if we look at our problem, |\alpha|^2 is the sum of the squares of two coordinates of a uniform point on S^3 (since SU(2) is isomorphic to S^3), hence by previous beta gamma relation, must be a beta(1,1) variable. Now recall the density of beta(a,b) is given by x^{a-1}(1-x)^{b-1}, so for us it would be constant! That’s why the distribution of M_1 under uniformly unitarily conjugated M_2 is in fact uniform in its admissible domain.

I couldn’t figure out the connection with Archimede’s epitaph yet. So please enlighten me…


About aquazorcarson

math PhD at Stanford, studying probability
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