as follows. In Section \[metstate\] we prove that hypothesis (H) holds under certain constraints on the magnetic field $h$ and the minimal degree of the graph $d_\mathrm{min}$. Section \[althyp\] gives an alternative to hypothesis (H), which holds for a broader range parameters, yet still permits us to claim our bounds on the crossover time. In Section \[thmproofs\] we prove our upper and lower bounds on $\Gamma_n^\star$. Part of this proof depends on a *dynamical construction* of ${\mathrm{CM}}_n$. In Section \[tailprop\] we derive certain properties of this construction.

Proof of Theorem \[thm:hyp\] {#metstate}
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This section gives a proof of hypothesis (H). We start with the following remark about the configurations in $\Omega$.

[A natural isomorphism between configurations and subsets of vertices of the underlying graph $G=(V,E)$ comes from identifying $\xi\in\Omega$ with the set $\{v \in V\colon\,\xi(v)=+1\}$. With this in mind, we denote by $\overline{\xi}$ the configuration corresponding to the complement of this set: $\{v\in V\colon\,\xi(v)=-1\}$. Furthermore, for $\zeta,\sigma\in\Omega$ we denote by $E(\zeta,\sigma) \subseteq E$ the set of all unoriented edges $\{(v,w)\in E\colon\,\zeta(v)=\sigma(w)=+1\}$. The main use of the last definition will be for $\sigma=\overline{\zeta}$: $E(\zeta,\overline{\zeta})$ is the edge boundary of the set $\{v\in V\colon\,\zeta(v)=+1\}$.]{}

We next give two lemmas that will 