Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices

Abstract: We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ random matrices has growth $ρ$ is at least inverse quasi-polynomial: $Ω(\exp(-c \log² (ρ)\log(n)))$ for some constant $c > 0$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.