Phase transitions in the condition number distribution of Gaussian random matrices

We study the statistics of the condition number $\kappa=\lambda_{\mathrm{max}}/\lambda_{\mathrm{min}}$ (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 $\langle\kappa\rangle$ (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 $\langle\kappa\rangle$, 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.