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

### Isaac Pérez Castillo 1 Eytan Katzav 2 Pierpaolo Vivo 3, 4

#### Physical Review E, American Physical Society, 2014, 90, pp.050103

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

• 1. Departamento de Sistemas Complejos, Instituto de Física, UNAM
• 2. Racah Institute of Physics - Racah Institute of Physics
• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
• 4. Department of Mathematics, King's College London