Binomial Random Matroids

TL;DR

This paper investigates the probability that a random collection of k-subsets forms the bases of a matroid, analyzing phase transitions, structural properties, and implications for counting matroids and related combinatorial objects.

Contribution

It provides asymptotic probability results for random matroid bases, identifies conditions preventing matroid formation, and improves estimates for the number of matroids and sparse paving matroids.

Findings

Probability of random collection forming a matroid undergoes phase transition.

When a matroid occurs, it is typically a sparse paving matroid.

Improved estimates for the number of matroids, paving matroids, and sparse paving matroids.

Abstract

Let $\mathcal B=\mathcal B_{k,n,p}$ be a random collection of $k$-subsets of $[n]$ where each possible set is present independently with probability $p$. Let $\cal E_{\mathcal B}$ be the event that $\mathcal B$ defines the set of bases of a matroid. We prove that If $p= 1-\frac{c_n}{(k(n-k)\binom nk)^{1/2}}$ where $0\leq c_n\leq \infty$, then \[ \lim_{n\to\infty}\Pr[\cal E_{\cal B}\mid |\cal B|\geq2]=\begin{cases}1&c_n\to0.\\e^{-c^2/2}&c_n\to c.\\0&c_n\to \infty.\end{cases}\] In addition, we identify a condition preventing the occurence of $\cal E_{\cal B}$ and prove a hitting time version for the occurence of $\cal B$. We also prove that when $\cal E_{\mathcal B}$ occurs, $\mathcal B$ defines a sparse paving matroid w.h.p. In addition, study a greedy algorithm that produces a random matroid defined by a collection of hyperplanes. We use this to improve the estimates in \cite{HPV} on $\log m(n,k),\log p(n,k), \log s(n,k)$ where $ m(n, k), p(n, k), s(n, k)$ denote the number of matroids, paving matroids, and sparse paving matroids (respectively) of rank $k$ on $[n]$. Our improvement lies in that we can deal with $k$ growing slowly with $n$ as opposed to $k=O(1)$ in \cite{HPV}. More generally, we obtain estimates for the number of matchings in nearly-regular hypergraphs with small codegree, which may be of independent interest.