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 \[ M_{i,j} = \langle σ_{a_i} σ_{b_j} \rangle \] 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.

20 pages, 3 figures

References in corpus (3)

• Total positivity, Grassmannians, and networks
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• A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field

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