proceed with the proof of hypothesis (H). Let $\sigma\in\Omega$ be any configuration that satisfies $x=\ell_{\sigma}/\ell_n\leq\tfrac12$, where $\ell_\sigma
=\sum_{i\in\sigma} d_i$. We will construct a path from $\sigma$ to some $\sigma\prime
\in\Omega$ satisfying $\mathcal{H}\left(\sigma\prime\right)<\mathcal{H}\left(\sigma\right)$ by removing one vertex at a time, obtaining a path $\sigma=\sigma_{0},\ldots,\sigma_{m}
=\sigma\prime$. In particular, at step $t$ we remove any vertex $v_{t}\in\sigma_{t-1}$ that minimises the quantity $\left|E\left(v_{t},\sigma_{t-1}\backslash v_{t}\right)\right|
-\left|E\left(v_{t},\overline{\sigma_{t-1}}\right)\right|$. It will follow that for every $\sigma_{i}$ in this path, we have $\left|\mathcal{H}\left(\sigma_{i}\right)-\mathcal{H}\left(\sigma_{0}
\right)\right|<\Gamma^{\star}$, which proves the claim of the theorem.

The probability that some configuration $\sigma$, chosen uniformly from all configurations in $\Omega$ with $\ell_\sigma=L$, has a boundary of size $\left|E\left(\sigma,
\overline{\sigma}\right)\right|=K$ equals $$\begin{aligned}
&{L \choose K}K!\left(L-K-1\right)!!{\ell_{n}-L \choose K}
\left(\ell_{n}-L-K-1\right)!!/\left(\ell_{n}-1\right)!! \\
&\approx \frac{\left(L\right)!}{\left(L-K\right)!!}
\frac{\left(\ell_{n}-L\right)!}{K!\left(\ell_{n}-L-K\right)!!}\frac{1}{\ell_{n}!!} \\
&\approx  L^L\left(\ell_{n}-L\right)^{\left(\ell_{n}-L\right)}
\left(\ell_{n}-L-K\right)^{-\left(\ell_{n}-L-K\right)/2}
\left(L-K\right)^{-\left(L-K\right)/2}K^{-K}\ell_{n}^{-\ell_{n}/2},
\end{aligned}$$ where the symbol $\approx$ stands for equality up to polynomial terms (here of order $O(n^{2})$). Let $x=L/\ell_{n}$ and $y=K/\ell_{n}$, so that the above expression becomes $$\exp\left[\ell_{n}\log\left(x{}^{x}\left(1-x\right)^{\left(1-x\right)}
\left(1-x-y\right)^{-\left(1-x-y\right)/2}\left(x-y\right)^{-\left(x-y\right)/2}y^{-y}\right)\right].$$ Furthermore, if we define $\eta\left(x\right)$ by $$\label{eq:defeta}
\exp\left[\ell_{n}\log\eta\left(x\right)\right]
=\left|\left\{ U\subseteq V\colon\,\ell_U=\ell_{n}x\right\} \right|,$$ then the probability of there being any configuration of total degree $L$ having a boundary size $K$ is bounded from above by $$\label{sd-expbound}
\exp\left[\ell_{n}\log\left(\eta\left(x\right)x{}^{x}\left(1-x\right)^{\left(1-x\right)}
\left(1-x-y\right)^{-\left(1-x-y\right)/2}\left(x-y\right)^{-\left(x-y\right)/2}y^{-y}\right)\right].$$

It is easy to see that, by using $\delta=d_{\min}$, the cardinality in the right-hand 