= Solution
Let $\vartheta$ be either permitted torus reflection and let $\Lambda_+$ be one reflected half. Under the product measure $\mu^{\otimes\Lambda_L}$, variables in the two open halves are independent and corresponding variables have the same law. For any $F$ depending on $\Lambda_+$,
$$
\int F\,\vartheta F\,d\mu^{\otimes\Lambda_L}
=\left|\int F\,d\mu^{\otimes\Lambda_+}\right|^2\geq0
$$
when the reflection has no fixed sites. If it fixes a layer of sites, condition on that layer; the same factorization gives a conditional square, whose expectation is nonnegative. This proves <reflection positivity> through sites. For an edge reflection the parity assumption makes the two halves pair exactly, so the first factorization applies there as well.
Back to article page
