Block dynamics comparison

Let $G$ be a graph with maximum degree $\Delta$ on a finite vertex set $V$, and let $\pi$ be the Ising distribution at inverse temperature $\beta>0$: $\pi(\sigma)\propto\exp\bigl(\beta\sum_{\{v,w\}\in E}\sigma(v)\sigma(w)\bigr)$ on spin configurations $\sigma:V\to\{\pm1\}$. Fix blocks $V_1,\dots,V_b\subseteq V$ covering $V$, each of size at most $M$, with every vertex lying in at most $M_\star$ blocks. The block dynamics picks a uniform block and re-samples the configuration on it from $\pi$ conditioned on the configuration outside; the single-site Glauber dynamics is the special case of singleton blocks. For either chain, the spectral gap is $\gamma=1-\lambda_2$ with $\lambda_2$ the largest eigenvalue different from $1$ (eigenvalues in the real-eigenvector sense of Mission VII); write $\gamma_B$ for the block dynamics' gap and $\gamma$ for the single-site gap.

The theorem (Theorem 15.9 of Levin–Peres–Wilmer) asserts the comparison

A spectral gap for the (coarse, easy-to-analyze) block dynamics transfers to the single-site dynamics at a cost depending only on the block size, the overlap multiplicity, the degree, and the temperature — not on the number of vertices. This is the engine of divide-and-conquer gap proofs: the book uses it for the tree theorem of this mission, and it is the template for gap bounds on lattices via recursive block decompositions. The proof routes each block update through a canonical path of single-site updates and bounds the congestion via the comparison method of Mission VII.