Phase transitions in the condition-number distribution of Gaussian random matrices

We study the statistics of the condition number κ=λmax/λmin (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[-βN2Φ-(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.