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 $a_1$,..., $a_k$ , $b_k$ ,..., $b_1$ 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 [37].