the empirical spectral distribution

Random matrix theory asks a single deceptively simple question: if you fill a large matrix with random numbers, what do its eigenvalues look like? An N-by-N matrix has N eigenvalues, and as N grows these N points scatter across the real line (for Hermitian matrices) or the complex plane (otherwise). To talk about their collective behaviour you need to package all N of them into one object, and that object is the empirical spectral distribution. It is the bridge between linear algebra (a list of eigenvalues) and probability (a random measure whose limit we can study).

Precisely, if a Hermitian N-by-N matrix A_N has eigenvalues lambda_1, ..., lambda_N, its empirical spectral distribution is the random probability measure mu_{A_N} = (1/N) sum over i of delta_{lambda_i}, the average of point masses placed at each eigenvalue. It puts mass 1/N on every eigenvalue, so it is a genuine probability measure on R, but a random one: it depends on the random entries of A_N. The whole game of bulk random matrix theory is to show that, after a suitable scaling of the entries (typically dividing by sqrt(N) so the eigenvalues stay O(1)), this random measure mu_{A_N} converges — usually in probability or almost surely, in the sense of weak convergence — to a deterministic limit measure as N tends to infinity. The limit is the limiting spectral distribution.

What makes this remarkable is that the limit is non-random even though every entry is random: the eigenvalues self-average, a manifestation of concentration in high dimensions. For Wigner matrices the limit is the semicircle law; for sample covariance matrices it is the Marchenko-Pastur law; for iid non-Hermitian matrices the (complex) limit is the uniform measure on a disc, the circular law. A crucial caveat: weak convergence of the empirical spectral distribution describes the bulk density only. It says nothing about the very largest or smallest eigenvalue, which can sit slightly outside the support of the limit and is governed by entirely separate edge theory (Tracy-Widom). Convergence of the measure does not imply convergence of extreme eigenvalues.