Edge removal perturbs the spectral gap by at most $e^{2\beta(\Delta+2r)}$

Let $G$ be a graph with maximum degree $\Delta$ on a finite vertex set, and let $G'\subseteq G$ be a subgraph obtained by deleting $r$ edges (all vertices kept). For a graph $H$, the Ising model at inverse temperature $\beta>0$ is the distribution $\pi_H(\sigma)\propto\exp\bigl(\beta\sum_{\{v,w\}\in E(H)}\sigma(v)\sigma(w)\bigr)$ on spin configurations $\sigma:V\to\{\pm1\}$, and its Glauber dynamics re-samples a uniformly chosen site from the conditional distribution. The spectral gap of a chain is $\gamma=1-\lambda_2$, where $\lambda_2$ is the largest eigenvalue different from $1$ (an eigenvalue being a real $\lambda$ with $Pf=\lambda f$ for some nonzero $f$, as in Mission VII).

The theorem (Proposition 15.7 of Levin–Peres–Wilmer) asserts that deleting the $r$ edges changes the spectral gap by at most an explicit exponential factor:

where $\gamma_H$ denotes the gap of the Glauber dynamics for the Ising model on $H$.

The proof is a direct comparison of Dirichlet forms (Mission VII): removing $r$ edges changes every Gibbs weight by at most $e^{2\beta r}$ and every transition probability by at most $e^{2\beta\Delta}$. In the book this is the key surgery step for the tree bound: cutting the root's edges splits the tree into independent subtrees, at a bounded cost in the gap.