On the asymptotic proportion of connected matroids

Very little is known about the asymptotic behavior of classes of matroids. We make a number of conjectures about such behaviors. For example, we conjecture that asymptotically almost every matroid: has a trivial automorphism group; is arbitrarily highly connected; and is not representable over any field. We prove one result: the proportion of labeled nn-element matroids that are connected is asymptotically at least 1/21/2.