still holds (see Dommers [@Dpr Lemma 5.3]). Thus, it follows from the arguments leading to that we only need the condition $$\frac{h}{J\ell_{n}}\left(\bar{m}+t+\left|\sigma\right|
-s\left(\frac{x-2I_{\delta}\left(x\right)}{x-I_{\delta}\left(x\right)}\right)\right)
\leq 2\frac{\ell_{\bar{m}}}{\ell_n}\Big(1-\frac{\ell_{\bar{m}}}{\ell_n}\Big)
-\left(x-2I_{\delta}\left(x\right)\right)^{2}\left(x-I_{\delta}\left(x\right)\right)^{-1}.
\label{eq:ihalfbound2}$$ For $h$ sufficiently small, the ratio $\frac{\ell_{\bar{m}}}{\ell_n}$ can be made arbitrarily close to $\tfrac12$, in which case the right-hand side of becomes strictly positive. This implies that the inequality in holds for any $\delta \geq 3$ whenever $h$ is sufficiently small.

Proof of Theorems \[thm:Ubound\] and \[thm:Lbound\] {#thmproofs}
===================================================

A dynamic construction of the configuration model {#CMdynamic}
-------------------------------------------------

Prior to giving the proof of Theorems \[thm:Lbound\] and \[thm:Ubound\], we introduce a *dynamical construction* of the CM graph. This will be used to obtain the upper bound in Theorem \[thm:Ubound\].

Let $V=\{v_{i}\} _{i=1}^{n}$ be a sequence of vertices with degrees $\{d_{i}\} _{i=1}^{n}$. In this section we construct a graph $G=(V,E)$ with the same distribution as a graph generated through the Configuration Model algorithm, but in a *dynamical way*, as follows.

Suppose that $\xi_{m}$ is a uniform random matching of the integers $\{1,\ldots,2m\} $, denoted by $\xi_{m}=\{(x_{1},x_{2}),\ldots,(x_{2m-1},
x_{2m}\}$, where the pairs are listed in the order they were created (which is not an important issue, so long as we agree on some labeling). Next, let 