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The Planar Ising Model and Total Positivity

Published version
Peer-reviewed

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Authors

Lis, M 

Abstract

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let a1,..., ak , bk ,..., b1 be vertices placed in a counterclockwise order on the outer face of G. We show that the k × k matrix of the two-point spin correlation functions

Mi,j = σaiσbj

is totally nonnegative. Moreover, det M > 0 if and only if there exist k pairwise vertex-disjoint paths that connect ai with bi . We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between ai and bi in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska [37].

Description

Keywords

Ising model, total positivity, random currents, alternating flows

Journal Title

Journal of Statistical Physics

Conference Name

Journal ISSN

0022-4715
1572-9613

Volume Title

166

Publisher

Springer
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
This research was supported by the Knut and Alice Wallenberg foundation, and was conducted when the author was at Chalmers University of Technology and the University of Gothenburg.