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Minors of matroids represented by sparse random matrices over finite fields

Pu Gao, Peter Nelson

Abstract

Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and $m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by determining the sharp threshold (of $m/n$) for the appearance of a fixed matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field.

Minors of matroids represented by sparse random matrices over finite fields

Abstract

Consider a random matrix over the finite field of order where every column has precisely nonzero elements, and let be the matroid represented by . In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid , if and where and are sufficiently large constants depending on N, then a.a.s. contains as a minor. We improve their result by determining the sharp threshold (of ) for the appearance of a fixed matroid as a minor of , for every , and every finite field.
Paper Structure (13 sections, 19 theorems, 65 equations, 2 figures)

This paper contains 13 sections, 19 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Other random matroid models
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:main']}(a) and Theorem \ref{['thm:main2']}(a)
  5. Proof of Theorem \ref{['thm:main']}(b) and Theorem \ref{['thm:main2']}(b)
  6. Completing the proofs of Theorems \ref{['thm:main']}(b) and \ref{['thm:main2']}(b)
  7. Proof of Lemmas \ref{['lem:rows']} and \ref{['lem:U']}
  8. Proof of Lemma \ref{['lem:minor']}
  9. Proof of Lemma \ref{['lem:minor']} when ${\mathbb F}={\mathbb F}_p$ where $p$ is prime
  10. Proof of Lemma \ref{['lem:minor']} when ${\mathbb F}={\mathbb F}_q$ where $q$ is a prime power
  11. Proof of the key lemma: Lemma \ref{['lem:key']}
  12. Proof of Lemma \ref{['lem:key']}
  13. Proof of Lemma \ref{['lem:corecounts']}.

Key Result

Theorem 2

Let $k\ge 3$. Suppose that ${\mathbb F}={\mathbb F}_p$ where $p$ a prime number, and let ${\mathcal{P}}$ be any permutation-invariant distribution on $({\mathbb F}_p^*)^k$. Let $N$ be a fixed simple ${\mathbb F}_p$-representable matroid. Then, for every fixed $\epsilon>0$ the following hold:

Figures (2)

  • Figure 1: The figure on the left is the Tanner graph of a pseudo-tree $T$; the figure on the right is the Tanner graph of $P(T)$.
  • Figure 2: An example of a hypergraph, its Tanner graph, and the 2-core of its Tanner graph

Theorems & Definitions (35)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Conjecture 6
  • Conjecture 7
  • Remark 8
  • Theorem 9
  • Proposition 10
  • ...and 25 more