2014/03/18 by Wood, Philip Matchett · 1 citation
#60B20 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1403.4593
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.