Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 2 / a / Solution

One step of the Glauber dynamics chooses a vertex $v$ uniformly and resamples its spin from its conditional Ising model distribution given all other spins. If the proposed new spin is $x$ and $S_v(\sigma )={\sum }_{w\sim v}\sigma (w)$, its update probability is