JournalsjemsVol. 24, No. 4pp. 1335–1351

Correlation bounds for fields and matroids

  • June Huh

    Princeton University, USA
  • Benjamin Schröter

    KTH Royal Institute of Technology, Stockholm, Sweden
  • Botong Wang

    University of Wisconsin-Madison, USA
Correlation bounds for fields and matroids cover
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Abstract

Let G be a finite connected graph, and let T be a spanning tree of G chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events e1Te_1 \in\rm T and e2Te_2 \in\rm T are negatively correlated for any distinct edges e1e_1 and e2e_2. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events eBe \in\rm B, where B is a randomly chosen basis of a matroid. As an application, we prove Mason’s conjecture that the number of kk-element independent sets of a matroid forms an ultra-log-concave sequence in kk.

Cite this article

June Huh, Benjamin Schröter, Botong Wang, Correlation bounds for fields and matroids. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1335–1351

DOI 10.4171/JEMS/1119