On the condition number of the critically-scaled laguerre unitary ensemble

Abstract

We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX∗, where X is an Nxn matrix with iid standard complex normal entries. Under the scaling n = N+ √4cN®90, c > 0 and N → ∞, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices A are all given by the Tracy-Widom distribution (β = 2). This scaling is motivated by the study of the solution of the equation Ax = b using the conjugate gradient algorithm, in the case that A and b are random: For such a scaling the fluctuations of the halting time for the algorithm are empirically seen to be universal.