hold is $$\tfrac{h}{J}\left(\tfrac{1}{d_{\mathrm{ave}}} + \tfrac{1}{2} \right)
< 2I_{d_{\mathrm{ave}}}\left(\tfrac{1}{2}\right)
-\tfrac{1}{2}\left(1-4I_{d_{\mathrm{min}}}\left(\tfrac{1}{2}\right)\right)^{2}
\left(1-2I_{d_{\mathrm{min}}}\left(\tfrac{1}{2}\right)\right)^{-1}.
\label{eq:Hcond1}$$ Hence we have a path $\sigma \to \sigma_{s}$ (or, when such an $s$ satisfying does not exist, a path $\sigma \to \boxminus$) with $\mathcal{H}\left(\sigma_{s}\right)
< \mathcal{H}\left(\sigma\right)$ that never exceeds $\mathcal{H}\left(\sigma\right)$ by $\Gamma^{\star}$ or more, whenever $h$ is sufficiently small and holds. This proves the claim of the theorem for all configurations $\sigma$ with $\ell_{\sigma}
\leq \tfrac12\ell_n$.

Note also that, for $\ell_{\sigma} > \tfrac12\ell_n$, the same argument can be repeated by adding a vertex at each step, which will also come at a lower cost since at each step the magnetisation changes by $-h$.

An alternative to hypothesis (H) {#althyp}
================================

In this section gives a weaker version of hypothesis (H), which nonetheless suffices as a prerequisite for Theorem \[thm:nucltime\]. This weaker version can be verified for a parameter range that is larger than the one needed in Section \[metstate\].

We can repeat the arguments given in Section \[metstate\]. But, instead of insisting that $\mathscr{V}_{\sigma}<\Gamma^{\star}$ for every configuration $\sigma \in \Omega$, we require that $\mathscr{V}_{\sigma}$ is bounded from above by our upper bound on $\Gamma^{\star}$, since this guarantees that our upper bound on the crossover time is still valid and 