\LinkDens \BoltCons T}{2 \Volu^n \LinkPote_0}
             \sum_{\REPP}\DensFunc_{\REPP}\DensFunc_{-\REPP}\LinkPoteRepl_{\REPP}\nonumber\\
        &&+\frac{\ExclVolu \PartNumb^2}{2\Volu_0} \sum_{p}\DensFunc_{p\BASE^{0}}
                \DensFunc_{-p\BASE^{0}} \nonumber\\
        &&+\frac{\ExclVolu \PartNumb^2}{2\Volu} \sum_{p} \ReplSumOne
                \DensFunc_{p\BASE\REPa} \DensFunc_{-p\BASE\REPa} ,\end{aligned}$$ where the factor $$\begin{aligned}
    \LinkPoteRepl_{\REPP} = \big(\LinkPote_0\big)^{1+n}
    e^{-\LinkScal^2 \vert \REPP \vert^2/2}\end{aligned}$$ is the replicated version of the Fourier transform of the function $\LinkPote(x)$, defined in Eq. (\[EQ:DeltReal\]). [[The summation $\sum_{\REPP}$ denotes a summation over all momentum $\Dime$-vectors $p\REPa$, one for each replica, with $\REPP$ taking the values $\REPP=\ReplSum p\REPa \BASE\REPa$. The cartesian components of the $p\REPa$ take the values $2\pi m/L$, where $L$ is the linear size of the system and $m$ is any integer. Similarly, summations $\sum_{p}$ over $\Dime$-vectors $p$ include components having the values $2\pi m/L$.]{}]{} The first term on the right hand side of Eq. (\[EQ:HQ\]) arises from the attractive, link-originating, interaction part \[see Eq. (\[EQ:ReplPart\])\]; the next two terms represent the excluded-volume interaction in $\HamiExcl$, for the $0^{\textrm{th}}$ replica and for replicas $1$ through $n$, respectively.

The excluded-volume interaction is taken to be very strong, and thus the density fluctuations in any single replica are heavily suppressed. This means that $\DensFunc_{p\REPa\BASE\REPa}$ are very small for all $p\REPa\ne 0$, and $\DensFunc_{p\REPa\BASE\REPa}\vert_{p\REPa=0}=1$, corresponding to a nearly homogeneous particle density. To manage this issue, we separate the replicated space into a 