The Planar Ising Model and Total Positivity

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].