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

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.