The limit of the smallest singular value of random matrices with i.i.d. entries

TL;DR

This paper proves that the scaled smallest singular value of certain random matrices with i.i.d. entries converges almost surely to a specific limit, under minimal moment conditions, resolving a longstanding open problem.

Contribution

It establishes almost sure convergence of the scaled smallest singular value under only second-moment assumptions, relaxing previous stronger moment conditions.

Findings

Convergence of scaled smallest singular value to 1−√z almost surely.

Minimal second-moment assumptions are sufficient for convergence.

Addresses a long-standing open problem in random matrix theory.

Abstract

Let $\{a_{ij}\}$ $(1\le i,j<\infty)$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m)_{m=1}^\infty$ satisfy $m/N_m\longrightarrow z$ for some $z\in(0,1)$. For each $m\in{\mathbb N}$ denote by $A_m$ the $N_m\times m$ random matrix $(a_{ij})$ $(1\le i\le N_m,1\le j\le m)$ and let $s_{m}(A_m)$ be its smallest singular value. We prove that the sequence $\bigl({N_m}^{-1/2} s_{m}(A_m)\bigr)_{m=1}^\infty$ converges to $1-\sqrt{z}$ almost surely. Our result does not require boundedness of any moments of $a_{ij}$'s higher than the $2$-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.