The above ar plots of the scaled level
densities of the Gaussian orthogonal ensemble of random matrices (beta
= 2) of sizes n = 4,6,8, and 10. They are plotted against the
scaled semicircle, which is the limiting distribution as n goes to
infinity. Notice the "bumps" in each picture! There are exactly as many
as the matrix size.
Now observe the level densities above; they
correspond to the size 4 Hermite ensembles of beta = 2 through 10. The
size of the 4 "bumps" increases with beta. The smallest-bumped graph
also appears in the subwindow 1 above. Can you guess what happens as we
let beta go to infinity? The answer is appears below in
background-colored ink (highlight it to read it).
distribution is the average of 4 Delta Dirac distributions. Care to
guess the supports? =)
The plot above was obtained with MOPs.