  We study the empirical spectral distribution (ESD) in the limit where n goes
to infinity of a fixed n by n matrix M_n plus small random noise of the form
f(n)X_n, where X_n has iid mean 0, variance 1/n entries and f(n) goes to 0 as n
goes to infinity. It is known for certain M_n, in the case where X_n is iid
complex Gaussian, that the limiting distribution of the ESD of M_n+f(n)X_n can
be dramatically different from that for M_n. We prove a general universality
result showing, with some conditions on M_n and f(n), that the limiting
distribution of the ESD does not depend on the type of distribution used for
the random entries of X_n. We use the universality result to exactly compute
the limiting ESD for two families where it was not previously known. The proof
of the main result incorporates the Tao-Vu replacement principle and a version
of the Lindeberg replacement strategy, along with the newly-defined notion of
stability of sets of rows of a matrix.
