18.338J/16.394J: The Mathematics of Infinite Random Matrices The Stieltjes transform based approach

As defined, the e.d.f. is right continuous and possibly atomic i.e. with step discontinuities at discrete points. In practical terms, the derivative of (1), referred to as the (eigenvalue) level density, is simply the appropriately normalized histogram of the eigenvalues of AN . The MATLAB code histn we distributed earlier approximates this density. A surprising result in infinite RMT is that for some matrix ensembles, the expectation E[FN (x)] has a well defined i.e. not zero and not infinite limit. We drop the notational dependence on N in (1) by defining the limiting e.d.f. as F(x) = lim N→∞ E[FN (x)]. (2)