### 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.

Original language | English |
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Pages (from-to) | 150-168 |

Number of pages | 19 |

Journal | European Journal of Combinatorics |

Volume | 59 |

DOIs | |

Publication status | Published - 1 Jan 2017 |

## Cite this

*European Journal of Combinatorics*,

*59*, 150-168. https://doi.org/10.1016/j.ejc.2016.08.005