Distribution of the Smallest Eigenvalue in the Correlated Wishart Model

TL;DR

This paper derives exact formulas for the distribution of the smallest eigenvalue in correlated Wishart matrices, providing new universality results and simplifying previous complex calculations in random matrix theory.

Contribution

It introduces a novel approach using matrix model dualities to compute the smallest eigenvalue distribution for fully correlated Wishart ensembles, avoiding complex integrals.

Findings

Exact distribution formulas for complex and real Wishart models.

New universality results on the local scale of the smallest eigenvalue.

Simplified expressions compared to previous methods.

Abstract

Wishart random matrix theory is of major importance for the analysis of correlated time series. The distribution of the smallest eigenvalue for Wishart correlation matrices is particularly interesting in many applications. In the complex and in the real case, we calculate it exactly for arbitrary empirical eigenvalues, i.e., for fully correlated Gaussian Wishart ensembles. To this end, we derive certain dualities of matrix models in ordinary space. We thereby completely avoid the otherwise unsurmountable problem of computing a highly non-trivial group integral. Our results are compact and much easier to handle than previous ones. Furthermore, we obtain a new universality for the distribution of the smallest eigenvalue on the proper local scale.