The planar Ising model and total positivity

TL;DR

This paper demonstrates that the matrix of two-point spin correlations in the planar Ising model with free boundary conditions is totally nonnegative, linking its positivity to the existence of disjoint paths, and explores its critical scaling limits.

Contribution

It establishes total nonnegativity of the correlation matrix in the planar Ising model and relates positivity to disjoint path existence, introducing new distributional relations involving random currents and flows.

Findings

Correlation matrix is totally nonnegative.

Determinant positivity linked to disjoint paths.

Scaling limit of connection probabilities at criticality.

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.