Phase transitions in the condition-number distribution of Gaussian random matrices
Por:
Castillo, IP, Katzav E., Vivo P.
Publicada:
26 nov 2014
Resumen:
We study the statistics of the condition number ?=?max/?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(?x) distributions of ?. We find that these distributions decay as P(?x)˜exp[-ßNf+(x)], where ß is the Dyson index of the ensemble. The left and right rate functions f±(x) are independent of ß and calculated exactly for any choice of the rectangularity parameter a=M/N-1>0. Interestingly, they show a weak nonanalytic behavior at their minimum '©?' (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 '©?', we determine exactly the scale of typical fluctuations ~O(N-2/3) and the tails of the limiting distribution of ?. The analytical results are in excellent agreement with numerical simulations. © 2014 American Physical Society.
Filiaciones:
Castillo, IP:
Univ Nacl Autonoma Mexico, Inst Fis, Dept Sistemas Complejos, Mexico City 01000, DF, Mexico
Katzav E.:
Racah Institute of Physics, Hebrew University, Jerusalem, 91904, Israel
Vivo P.:
Department of Mathematics, King's College London, Strand, London, WC2R 2LS, United Kingdom
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