Evolution of random representable matroids: minors, circuits, connectivity and the critical number

TL;DR

This paper investigates the probabilistic evolution of random matroids generated by sequentially adding random vectors over finite fields, analyzing properties like minors, circuits, connectivity, and the critical number.

Contribution

It provides new insights into the evolution of random matroids, resolving several open problems related to their structural properties.

Findings

Characterization of the emergence of minors and circuits in random matroids

Analysis of connectivity evolution in the process

Determination of the critical number behavior over the sequence

Abstract

We study the evolution of random matroids represented by the sequence of random matrices over ${\mathbb F}_q$ where columns are added one after the other, and each column vector is a uniformly random vector in ${\mathbb F}_q^n$, independent of each other. We study the appearance of matroid minors, the appearance of circuits, the evolution of the connectivities and the critical number. We settle several open problems in the literature.