The asymptotic distribution of the condition number for random circulant matrices

TL;DR

This paper investigates the asymptotic behavior of the condition number for large random circulant matrices, showing it converges to a Fréchet distribution under certain conditions.

Contribution

It provides the first rigorous analysis of the joint distribution of extremal singular values for random circulant matrices and establishes their limiting laws.

Findings

Joint law of extremal singular values converges to independent Rayleigh and Gumbel laws.

Normalized condition number converges to a Fréchet distribution.

Results hold for matrices with i.i.d. entries satisfying Lyapunov condition.

Abstract

In this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized condition number converges in distribution to a Fr\'echet law as the dimension of the matrix increases.