The Planar Ising Model and Total Positivity

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 (a_1,dots ,a_k,b_k,dots ,b_1) be vertices placed in a counterclockwise order on the outer face of G. We show that the (ktimes k) matrix of the two-point spin correlation functions

$$begin{aligned} M_{i,j} = langle sigma _{a_i} sigma _{b_j} rangle end{aligned}$$

is totally nonnegative. Moreover, (det M > 0) if and only if there exist k pairwise vertex-disjoint paths that connect (a_i) with (b_i). We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between (a_i) and (b_i) 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].