  The model theory based notion of the first order convergence unifies the
notions of the left-convergence for dense structures and the Benjamini-Schramm
convergence for sparse structures. It is known that every first order
convergent sequence of graphs with bounded tree-depth can be represented by an
analytic limit object called a limit modeling. We establish the matroid
counterpart of this result: every first order convergent sequence of matroids
with bounded branch-depth representable over a fixed finite field has a limit
modeling, i.e., there exists an infinite matroid with the elements forming a
probability space that has asymptotically the same first order properties. We
show that neither of the bounded branch-depth assumption nor the
representability assumption can be removed.
