First order convergence of matroids [article]

Frantisek Kardos and Daniel Kral and Anita Liebenau and Lukas Mach
<span title="2016-08-13">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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
more &raquo; ... xed 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.
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