title: Phase transitions in the condition-number distribution of Gaussian random matrices.
abstract: We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and smallest squared singular values) of N×M Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large N the cumulative P(κ<x) and tail-cumulative P(κ>x) distributions of κ. We find that these distributions decay as P(κ<x)≈exp[-βN^{2}Φ_{-}(x)] and P(κ>x)≈exp[-βNΦ_{+}(x)], where β is the Dyson index of the ensemble. The left and right rate functions Φ_{±}(x) are independent of β and calculated exactly for any choice of the rectangularity parameter α=M/N-1>0. Interestingly, they show a weak nonanalytic behavior at their minimum 〈κ〉 (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around 〈κ〉, we determine exactly the scale of typical fluctuations ∼O(N^{-2/3}) and the tails of the limiting distribution of κ. The analytical results are in excellent agreement with numerical simulations.
authors.affiliation: Departamento de Sistemas Complejos, Instituto de Física, UNAM, P.O. Box 20-364, 01000 México Distrito Federal, Mexico.
authors.forename: Isaac
authors.initials: I
authors.lastname: Pérez Castillo
authors.affiliation: Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel.
authors.forename: Eytan
authors.initials: E
authors.lastname: Katzav
authors.affiliation: Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom.
authors.forename: Pierpaolo
authors.initials: P
authors.lastname: Vivo