(15/5,30\*0.0825) (16/5,30\*0.0844) (17/5,30\*0.0854) (18/5,30\*0.0864) (19/5,30\*0.0869) (20/5,30\*0.0870)]{};

\[thm:Lbound\] Define $$\label{eq-defmtilde}
\tilde{m} = \min\left\lbrace 1 \leq m \leq n\colon\, \ell_m \geq \tfrac12\ell_n\right\rbrace.$$ Then, w.h.p., $$\Gamma_n^\star \geq \Gamma_n^-, \qquad
\Gamma_n^- = J\,d_\mathrm{ave}\,I_{d_\mathrm{ave}}
\left(\tfrac12\right) n - h\tilde{m} - o(n).
\label{eq:lboundGamma}$$

Under hypothesis [(H)]{} (or the weaker version of [(H)]{} introduced in Section \[althyp\]), Theorems [\[thm:Ubound\]–\[thm:Lbound\]]{} yield the following bounds on the crossover time (see Dommers [@Dpr Proposition 2.4]): $$\label{timesandwich}
\lim_{\beta\to\infty} P_{\boxminus}^{G,\beta}\left(e^{\Gamma_n^- -\varepsilon}
\leq \tau_\boxplus \leq e^{\Gamma_n^+ +\varepsilon}\right) =1.$$

In Corollary \[cor:mvalues\] we compute $\bar{m}$, $\ell_{\bar{m}}$, $\tilde{m}$ for two degree distributions: Dirac distributions and power-law distributions. It is clear that $\tilde{m}=\lceil \tfrac12 n\rceil$ for Dirac distributions.

The bounds we have found in Theorems \[thm:Ubound\]–\[thm:Lbound\] are tight in the limit of large degrees. Indeed, by the law of large numbers we have that $$\ell_{n}\frac{\ell_{\bar{m}}}{\ell_{n}}
\left(1-\frac{\ell_{\bar{m}}}{\ell_{n}}\right)
\leq\tfrac{1}{4}\ell_{n}=\tfrac{1}{4} d_{\mathrm{ave}}\,n\left[1+o(1)\right].$$ Hence $$\frac{\Gamma_n^+}{\Gamma_n^-}
= \frac{\frac{1}{4}d_{\mathrm{ave}}\left[1+o\left(1\right)\right]
-\frac{h}{J}\frac{\bar{m}}{n}+o(1)}{d_{\mathrm{ave}}
I_{d_{\mathrm{ave}}}\left(\frac{1}{2}\right)-\frac{h}{J}\frac{\tilde{m}}{n}
-o(1)}.
\label{eq:ratioofbounds}$$ In the limit as $d_{\mathrm{ave}}\to\infty$ we have $I_{d_{\mathrm{ave}}}
\left(\frac{1}{2}\right)\to\frac{1}{4}$, in which case tends to 1.

Discussion
----------

We close this introduction by discussing our main results.

[**1.**]{} We believe that Theorem \[thm:hyp\] holds as soon as $$0 < h < (d_\mathrm{min}-1)J,$$ i.e., we believe that in the limit as $\beta\to\infty$ followed by $n\to\infty$ this choice of parameters corresponds to the *metastable 