Phase transitions in the condition number distribution of Gaussian random matrices

Abstract

We study the statistics of the condition number $\kappa ={\lambda }_{\max }/{\lambda }_{\min }$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\mathcal{P}[\kappa <x]$ and tail-cumulative $\mathcal{P}[\kappa >x]$ distributions of $\kappa$. We find that these distributions decay as $\mathcal{P}[\kappa <x]\approx \exp \left(-\beta N^2{\Phi }_{-}(x)\right)$ and $\mathcal{P}[\kappa >x]\approx \exp \left(-\beta N{\Phi }_{+}(x)\right)$, where $\beta$ is the Dyson index of the ensemble. The left and right rate functions ${\Phi }_{\pm }(x)$ are independent of $\beta$ and calculated exactly for any choice of the rectangularity parameter $\alpha =M/N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $⟨\kappa ⟩$ (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 $⟨\kappa ⟩$, we determine exactly the scale of typical fluctuations $\sim \mathcal{O}(N^{-2/3})$ and the tails of the limiting distribution of $\kappa$. The analytical results are in excellent agreement with numerical simulations.

arXiv comment5 pag. + 7 pag. Suppl. Material. 3 Figures

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