Inclusion of forbidden minors in random representable matroids

In 1984, Kelly and Oxley introduced the model of a random representable matroid M[A(n)] corresponding to a random matrix A(n) is an element of F-q(m(n)xn), whose entries are drawn independently and uniformly from 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 -> infinity that a fixed F-q-representable matroid M is a minor of M[A(n)]. (We always assume m(n) >= 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) -> infinity, but is a.a.s. not if m(n) - n -> infinity. 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[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 M of F-q-representable matroids, provided: (i) n - m(n) -> infinity, and (ii) m(n) is at least the minimum rank of any F-q-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. (C) 2017 Elsevier B.V. All rights reserved.