In 1984, Kelly and Oxley introduced the model of a random representable matroid M [A n ] corresponding to a random matrix A n ∈ F m(n)×n q , 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 → ∞ that a fixedfor 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) → ∞, 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 probability that M is a minor ofThe 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) → ∞, 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.
We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders.In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [JST + 04].
In prior work, Cho and Kim studied competition graphs arising from doubly partial orders. In this article, we consider a related problem where competition graphs are instead induced by permutations. We first show that this approach produces the same class of competition graphs as the doubly partial order. In addition, we observe that the 123 and 132 patterns in a permutation induce the edges in the associated competition graph. We classify the competition graphs arising from 132-avoiding permutations and show that those graphs must avoid an induced path graph of length 3. Finally, we consider the weighted competition graph of permutations and give some initial enumerative and structural results in that setting.
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