Small deviations of determinants of random matrices with Gaussian entries

TL;DR

This paper estimates the probability of small deviations in the determinant of a Gaussian random matrix's product, providing sharp inequalities for these probabilities.

Contribution

It introduces precise bounds for small deviation probabilities of determinants of Gaussian random matrices, advancing understanding of their probabilistic behavior.

Findings

Derived sharp inequalities for small deviation probabilities

Provided asymptotic estimates for determinants of Gaussian matrices

Enhanced theoretical understanding of Gaussian matrix determinants

Abstract

The probability of the small deviations of the matrix $AA^T$ determinant is estimated, where $A$ is an $n\times\infty$ random matrix with centered entries having joint Gaussian distribution. The inequality obtained is sharp in a sence.