Inclusion of forbidden minors in random representable matroids

Abstract

In 1984, Kelly and Oxley introduced the model of a random representable matroid M[An] corresponding to a random matrix An∈Fqm(n)×n, whose entries are drawn independently and uniformly from Fq. 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→∞ that a fixed Fq-representable matroid M is a minor of M[An]. (We always assume m(n)≥rank(M) for all sufficiently large n, otherwise M can never be a minor of the corresponding M[An].) When M is free, we show that M is asymptotically almost surely (a.a.s.) a minor of M[An]. When M is not free, we show a phase transition: M is a.a.s. a minor if n−m(n)→∞, but is a.a.s. not if m(n)−n→∞. In the more general settings of m≤n and m>n, we give lower and upper bounds, respectively, on both the asymptotic and non-asymptotic probabilities that M is a minor of M[An]. The tools we develop to analyze matroid operations and minors of random matroids may be of independent interest.Our results directly imply that M[An] is a.a.s. not contained in any proper, minor-closed class M of Fq-representable matroids, provided: (i) n−m(n)→∞, and (ii) m(n) is at least the minimum rank of any Fq-representable forbidden minor of 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.