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 $A=X{X}^{\ast }$, where $X$ is an $N\times n$ matrix with iid standard complex normal entries. Under the scaling $n=N+\lfloor \sqrt{4cN}\rfloor$, $c>0$ and $N\to \infty$, 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.

References in corpus (2)

• Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble
• Universality in Numerical Computations with Random Data. Case Studies

Cited by in corpus (2)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Smoothed Analysis for the Conjugate Gradient Algorithm