Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 2 / iv / 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 ${\Lambda }_{+}$,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.