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

الأصل · EN

We study the empirical spectral distribution (ESD) in the limit where n goes to infinity of a fixed n by n matrix Mₙ plus small random noise of the form f(n)Xₙ, where Xₙ has iid mean 0, variance 1/n entries and f(n) goes to 0 as n goes to infinity. It is known for certain Mₙ, in the case where Xₙ is iid complex Gaussian, that the limiting distribution of the ESD of Mₙ+f(n)Xₙ can be dramatically different from that for Mₙ. We prove a general universality result showing, with some conditions on Mₙ and f(n), that the limiting distribution of the ESD does not depend on the type of distribution used for the random entries of Xₙ. 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.