tending to $1$ as $n\to\infty$ (see van der Hofstad [@vdHpr]).

Main theorems {#maintheorems}
-------------

We are interested in proving hypothesis (H) and identifying the key quantities in for $G = {\mathrm{CM}}_n$, which we henceforth denote by $({\mathcal{C}}^\star_n,\Gamma^\star_n,K^\star_n)$, in the limit as $n\to\infty$.

Our first main theorem settles hypothesis (H) for small magnetic field.

\[thm:hyp\] Suppose that the inequality in equation holds. Then $$\lim_{n\to\infty} {\mathbb{P}}_n\big({\mathrm{CM}}_n \text{ satisfies {\rm (H)}}\big) = 1.$$

Our second and third main theorem provide upper and lower bounds on $\Gamma^\star_n$. Label the vertices of the graph so that their degrees satisfy $d_{1}\leq\ldots\leq d_{n}$. Let $\gamma\colon\,\boxminus\to\boxplus$ be the path that successively flips the vertices $v_{1},\ldots,v_{n}$ (in that order), and let $\ell_{m}=\sum_{i=1}^{m}d_{i}$.

\[thm:Ubound\] Define $$\label{eq-defmbar}
\bar{m} = \min\left\{1 \leq m \leq n\colon\,\ell_m\left(1-\frac{\ell_m}{\ell_n}\right)
\geq \ell_{m+1}\left(1-\frac{\ell_{m+1}}{\ell_n}\right)-\frac{h}{J}\right\} < \frac{n}{2}.$$ Then, w.h.p., $$\Gamma_n^{\star} \leq \Gamma_n^+,
\qquad \Gamma_n^+ = J\ell_{\bar{m}}\Big(1-\frac{\ell_{\bar{m}}}{\ell_n}\Big)
-h\bar{m} \pm O\big(\ell_n^{3/4}\big).$$

For $0<x\leq\tfrac12$ and $\delta>1$, define (see Fig. \[fig:functionI\]) $$\begin{aligned}
I_{\delta}\left(x\right) &= \inf\Big\{0<y\leq x\colon\\
&1<x{}^{x\left(1-1/\delta\right)}
\left(1-x\right)^{\left(1-x\right)\left(1-1/\delta\right)}
\left(1-x-y\right)^{-\left(1-x-y\right)/2}\left(x-y\right)^{-\left(x-y\right)/2}y^{-y}\Big\}.
\end{aligned}
\label{eq:functionI}$$

(0,0) – (5,0) node\[anchor=north\] [$x$]{}; (0,0) – (0,4) node\[anchor=east\] [$I_{\delta}$]{}; (0,0) node\[anchor=north\] [0]{} (2,0) node\[anchor=north\] [$\tfrac{1}{4}$]{} (4,0) node\[anchor=north\] [$\tfrac{1}{2}$]{}; (0,1.4) node\[anchor=east\] [$\tfrac{11}{250}$]{} (0,2.8) node\[anchor=east\] [$\tfrac{11}{125}$]{}; plot \[smooth\] coordinates [(0,0) (1/5,30\*0.0111) (2/5,30\*0.0205) (3/5,30\*0.0286) (4/5,30\*0.0361) (5/5,30\*0.0429) (6/5,30\*0.0490) (7/5,30\*0.0546) (8/5,30\*0.0596) (9/5,30\*0.0641) (10/5,30\*0.0683) (11/5,30\*0.0721) (12/5,30\*0.0753) (13/5,30\*0.0780) (14/5,30\*0.0805) 