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 $a_1,\dots,a_k,b_k,\dots,b_1$ 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 [ M_{i,j} = \langle \sigma_{a_i} \sigma_{b_j} \rangle ] 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.