Inclusion of Forbidden Minors in Random Representable Matroids

TL;DR

This paper investigates the likelihood that fixed representable matroids appear as minors in large random matrices over finite fields, revealing phase transitions and implications for minor-closed classes.

Contribution

It provides the first asymptotic analysis of forbidden minors in random representable matroids, establishing phase transition thresholds and bounds for their occurrence.

Findings

When the matroid is free, it is almost surely a minor.

For non-free matroids, a phase transition depends on the relation between n and m(n).

Random matroids are almost surely not contained in any proper minor-closed class under certain conditions.

Abstract

In 1984, Kelly and Oxley introduced the model of a random representable matroid $M[A_n]$ corresponding to a random matrix $A_n \in \mathbb{F}_q^{m(n) \times n}$, whose entries are drawn independently and uniformly from $\mathbb{F}_q$. Whereas properties such as rank, connectivity, and circuit size have been well-studied, forbidden minors have not yet been analyzed. Here, we investigate the asymptotic probability as $n \to \infty$ that a fixed $\mathbb{F}_q$-representable matroid $M$ is a minor of $M[A_n]$. (We always assume $m(n) \geq \text{rank}(M)$ for all sufficiently large $n$, otherwise $M$ can never be a minor of the corresponding $M[A_n]$.) When $M$ is free, we show that $M$ is asymptotically almost surely (a.a.s.) a minor of $M[A_n]$. When $M$ is not free, we show a phase transition: $M$ is a.a.s. a minor if $n - m(n) \to \infty$, but is a.a.s. not if $m(n) - n \to \infty$. In the more general settings of $m \leq n$ and $m > n$, we give lower and upper bounds, respectively, on both the asymptotic and non-asymptotic probability that $M$ is a minor of $M[A_n]$. The tools we develop to analyze matroid operations and minors of random matroids may be of independent interest. Our results directly imply that $M[A_n]$ is a.a.s. not contained in any proper, minor-closed class $\mathcal{M}$ of $\mathbb{F}_q$-representable matroids, provided: (i) $n - m(n) \to \infty$, and (ii) $m(n)$ is at least the minimum rank of any $\mathbb{F}_q$-representable forbidden minor of $\mathcal{M}$, for all sufficiently large $n$. As an application, this shows that graphic matroids are a vanishing subset of linear matroids, in a sense made precise in the paper. Our results provide an approach for applying the rich theory around matroid minors to the less-studied field of random matroids.