Universality of the ESD for a fixed matrix plus small random noise: A stability approach

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.