from $\xi$ to $\zeta$ corresponds to a flip of a single spin, and is referred to as an *allowed move*. Glauber spin-flip dynamics on $\Omega$ is the continuous-time Markov process $(\xi_t)_{t \geq 0}$ defined by the transition rates $$c_\beta(\xi,\zeta) = \begin{cases}
e^{-\beta[{\mathcal{H}}(\zeta)-{\mathcal{H}}(\xi)]_+}, &\xi\sim\zeta,\\
0, &\mbox{otherwise}.
\end{cases}$$ The Gibbs measure in is the reversible equilibrium of this dynamics. We write $P^{G,\beta}_\xi$ to denote the law of $(\xi_t)_{t \geq 0}$ given $\xi_0=\xi$, ${\mathcal{L}}^{G,\beta}$ to denote the associated generator, and $\lambda^{G,\beta}$ to denote the principal eigenvalue of ${\mathcal{L}}^{G,\beta}$. The upper indices $G,\beta$ exhibit the dependence on the underlying graph $G$ and the interaction strength $\beta$ between neighbouring spins. For $A\subseteq\Omega$, we write $$\tau_A = \inf\big\{t>0\colon\,\xi_t \in A,\,\exists\,0<s<t\colon\,\xi_s \neq \xi_0\big\}$$ to denote the first hitting time of the set $A$ after the starting configuration is left.

Metastability
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To describe the metastable behaviour of our dynamics we need the following geometric definitions.

\(a) The communication height between two distinct configurations $\xi,\zeta\in\Omega$ is $$\Phi(\xi,\zeta) = \min_{\gamma\colon\,\xi\to\zeta} \max_{\sigma\in\gamma}
{\mathcal{H}}(\sigma),$$ where the minimum is taken over all paths $\gamma\colon\,\xi\to\zeta$ consisting of allowed moves only. The communication height between two non-empty disjoint sets $A,B\subset\Omega$ is $$\Phi(A,B) = \min_{\xi\in A,\zeta\in B} \Phi(\xi,\zeta).$$ (b) The stability level of $\xi\in\Omega$ is 