The joint gluing inequality in the language of Green's functions: An exposition of a proof by OpenAI

We re-present the proof of the joint gluing inequality for independent percolation from OpenAI's preprint "No percolation at criticality on quasi-transitive graphs" (Section 6), in the vocabulary of Markov-chain potential theory. The estimates and key identities are OpenAI's. We add only a change of language and a few elementary rearrangements, flagged as such, and claim no new theorem. Two Green functions organize the argument. In the statement, the triangular matrix of conditioned hitting probabilities is the Green (visit) matrix of a killed Markov chain on the list of relay vertices, and the gluing inequality says that a certain function of the starting relay is excessive for this chain. In the proof, the positivity of that chain follows from a Poisson equation for the two-cluster Gibbs sampler of van den Berg, Häggström and Kahn (2006), solved by the sampler's own Green function.