Minors of matroids represented by sparse random matrices over finite fields

TL;DR

This paper determines the precise threshold for when a fixed matroid appears as a minor in a sparse random matrix over finite fields, extending previous results from binary fields to all finite fields.

Contribution

It improves prior work by identifying the exact threshold for the appearance of a fixed matroid as a minor in sparse random matrices over any finite field.

Findings

Sharp threshold for matroid minors established

Results extend to all finite fields beyond binary case

Provides precise conditions for matroid minor appearance

Abstract

Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and $m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by determining the sharp threshold (of $m/n$) for the appearance of a fixed matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field.