we assume a version of the normalized link distribution as follows: $$\begin{aligned}
    \DEDist(\RealDiso)
    = \frac{
            \Big(\frac{\LinkDens \Volu_0}{2\PartNumb \LinkPote_0}\Big)^{\LinkNumb}
            Z_{\RealDiso}(\Volu_0)}
        {\LinkNumb ! Z_1},\end{aligned}$$ where $\LinkDens$ is a parameter that controls the mean total number of links. We assume that the preparation state (i.e., the state in which the links are going to be introduced) is in a given volume $\Volu_0$. The $Z_{\RealDiso}(\Volu_0)$ factor is actually the partition function, as given in Eq. (\[EQ:partition\]), and can be regarded as probing the equilibrium correlations of the underlying unlinked liquid. The factor $\LinkPote_0=\big(2\pi \LinkScal^2 \big)^{d/2}$ is actually the $p=0$ value of the Fourier transform of the $\LinkPote$ function defined in Eq. (\[EQ:DeltReal\]), and we shall see later that these factors ensure that the (mean-field) critical point occurs at $\LinkDens_C =1$. The normalization factor $Z_1$ is defined to be $\sum_{\RealDiso} \big(\frac{\LinkDens \Volu_0}{2\PartNumb \LinkPote_0}\big)^{\LinkNumb} Z_{\RealDiso}(\Volu_0)/ \LinkNumb !$. The calculation for $Z_1$ is straightforward, and is given in Appendix \[APP:DisoAver\].

The Deam-Edwards distribution can be understood as arising from a realistic vulcanization process in which the links are introduced simultaneously and instantaneously into the liquid state in equilibrium. Specifically, it incorporates the notion that all pairs of particles that happen (at some particular instant) to be nearby 