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Transfer current and pattern fields in spanning trees

Kassel, Adrien ; Wu, Wei

In: Probability Theory and Related Fields, 2015, vol. 163, no. 1-2, p. 89-121

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    Titre
    • Transfer current and pattern fields in spanning trees
    Auteur
    • Kassel, Adrien. Department Mathematik, ETH, Rämistrasse 101, 8092, Zurich, Switzerland
    • Wu, Wei. Division of Applied Mathematics, Brown University, 182 George Street, 02912, Providence, RI, USA
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Probability Theory and Related Fields, 2015, vol. 163, no. 1-2, p. 89-121. Springer Berlin Heidelberg
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:332232
    Summary
    When a simply connected domain $$D\subset {{\mathbb {R}}}^d$$ D ⊂ R d ( $$d\ge 2$$ d ≥ 2 ) is approximated in a "good” way by embedded connected weighted graphs, we prove that the transfer current matrix (defined on the edges of the graph viewed as an electrical network) converges, up to a local weight factor, to the differential of Green's function on $$D$$ D . This observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit. We further show that, on a periodic approximation of the domain, all pattern fields of the spanning tree model, as well as the minimal-pattern (e.g. zero-height) fields of the sandpile, converge weakly in distribution to Gaussian white noise.