First order convergence of matroids

TL;DR

This paper extends the concept of first order convergence from graphs to matroids, proving that sequences with bounded branch-depth and fixed finite field representability have a limit modeling, highlighting the limits of these conditions.

Contribution

It establishes the matroid analogue of first order convergence and demonstrates the necessity of bounded branch-depth and representability assumptions.

Findings

Every first order convergent sequence of matroids with bounded branch-depth has a limit modeling.

The bounded branch-depth and representability assumptions are necessary for the existence of a limit modeling.

The result unifies model theory concepts with matroid theory, extending convergence notions to combinatorial structures.

Abstract

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.