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