The smallest singular value of a shifted $d$-regular random square matrix

TL;DR

This paper establishes a probabilistic lower bound on the smallest singular value of a random $d$-regular matrix, which is the adjacency matrix of a directed graph with fixed degree, for a range of $d$ relative to matrix size.

Contribution

It provides the first non-trivial lower bound on the smallest singular value of shifted $d$-regular matrices within a specified degree range.

Findings

Smallest singular value exceeds $c_2 n^{-6}$ with high probability.

Probability bound improves as degree $d$ increases.

Results are independent of specific matrix entries, depending only on degree and size.

Abstract

We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, and $C_2$ are absolute positive constants independent of any other parameters.