Séminaire exceptionnel du LPTMS: Pierpaolo Vivo


14:30 - 15:30

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405
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Phase transitions in the condition number distribution of Gaussian random matrices

Pierpaolo Vivo, LPTMS

We study the statistics of the condition number k (the ratio between largest and smallest squared singular values) of NxM Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large N its cumulative P(k<x) and tail-cumulative P(k>x) distributions. We find the decay of these distribution. The left and right rate functions are calculated exactly for any choice of the rectangularity parameter M/N-1. Interestingly, they show a weak non-analytic behavior at their minimum <k> (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 <k>, we determine exactly the scale of typical fluctuations and the tails of the limiting distribution of  k. The analytical results are in excellent agreement with numerical simulations.

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