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

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

The worst-case distance to stationarity isThe Path coupling theorem says that if a coupling contracts an integer-valued path metric by a factor $\alpha <1$ for every adjacent pair, then

Couple two configurations differing at one vertex $u$ by choosing the same update vertex and the same uniform random number for the heat-bath update. Updating $u$ removes the disagreement. Updating a nonneighbor of $u$ cannot create one. At a neighbor $v$, the two conditional plus-spin probabilities differ by at most $\tanh \beta$, using the supplied identity. Since $u$ has at most $\Delta$ neighbors, the expected Hamming distance after one step is at mostThe Hamming diameter is $n$, so the Path coupling theorem gives