2015/07/02 by Percy Deift, Deift, Percy, Govind Menon +3
Computer Science · Mathematics · Physics and Astronomy · #35Q15 #60B20 #65C50 #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #FOS: Mathematics #Morphological variations and asymmetry #Probability (math.PR) #Random Matrices and Applications #Statistical Mechanics and Entropy #math.PR #msc:35Q15 #msc:60B20 #msc:65C50
paper · pdf · doi:10.48550/arxiv.1507.00750
openalex publication_date 2015/07/02 · arxiv created 2015/08/17 · arxiv updated 2015/08/19 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX^*, where X is an N × n matrix with iid standard complex normal entries. Under the scaling n = N + \lfloor √( 4 c N) \rfloor, 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.