for the latter. In Section \[discussion\] we place these results in their proper context and give an outline of the remainder of the paper.

Ising model and Glauber dynamics {#IsingGlauber}
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Given a finite connected non-oriented multigraph $G=(V,E)$, let $\Omega=\{-1,+1\}^V$ be the set of configurations $\xi=\{\xi(v)\colon\,v \in V\}$ that assign to each vertex $v\in V$ a spin-value $\xi(v)\in\{-1,+1\}$. Two configurations that will be of particular interest to us are those where all spins point up, respectively, down: $$\boxplus \equiv +1, \qquad \boxminus \equiv -1.$$ For $\beta \geq 0$, playing the role of *inverse temperature*, we define the Gibbs measure $$\mu_\beta(\xi) = \frac{1}{Z_\beta}\,e^{-\beta {\mathcal{H}}(\xi)},
\qquad \xi\in\Omega,
\label{eq:Gibbs}$$ where ${\mathcal{H}}\colon\,\Omega\to{\mathbb{R}}$ is the *Hamiltonian* that assigns an energy to each configuration given by $${\mathcal{H}}(\xi) = -\frac{J}{2} \sum_{(v,w) \in E} \xi(v)\xi(w)
-\frac{h}{2} \sum_{v \in V} \xi(v), \qquad \xi\in\Omega,
\label{eq:hamiltonian}$$ with $J>0$ the *ferromagnetic pair potential* and $h>0$ the *magnetic field*. The first sum in the right-hand side of runs over all non-oriented edges in $E$. Hence, if $v,w \in V$ have $k \in {\mathbb{N}}_0$ edges between them, then their joint contribution to the energy is $-k\,\frac{J}{2}\,\xi(v)\xi(w)$.

We write $\xi\sim\zeta$ if and only if $\xi$ and $\zeta$ agree at all but one vertex. A transition 