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 $k$ $\times$ $k$ matrix of the two-point spin correlation functions

$Mi,j$ = $⟨$$\sigma$$\sigma$$⟩$

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].