On the condition number of the critically-scaled Laguerre Unitary Ensemble

We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices $A=X{X}^{\ast }$, where $X$ is an $N\times n$ matrix with iid standard complex normal entries. Under the scaling $n = N + \lfloor \sqrt{ 4 c N} \rfloor$,$c > 0$and$N \rightarrow \infty$, 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 ($\beta =2$). 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.