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 \(k\times k\) 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].