constituent entities.

Equation (\[EQ:HRLPM\]) is a Hamiltonian for a given realization of quenched disorder $\RealDiso \equiv \{ i_e, j_e\}_{e=1}^{\LinkNumb}$, which describes the particular random instance of the linking of the particles. These links are the quenched disorder of the system, which are specified at synthesis and do not change with thermal fluctuations. This is because there is a wide separation between the timescale for the linked-particle system to reach thermal equilibrium and the much longer timescale required for the links themselves to break. Therefore, we treat the links as permanent. Later, we shall apply the replica technique [@Mezard1987] to average over these permanent random links.

Replica statistical mechanics of the RLPM {#SEC:RLPMReplica}
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For a given volume and a given realization of disorder $\RealDiso$ we can write the partition function $Z_{\RealDiso}$ for the RLPM as $$\begin{aligned}
\label{EQ:partition}
    Z_{\RealDiso} (\Volu)
    &\equiv& \int _{\Volu} \prod_{i=1}^{\PartNumb} d \PartPosi_i
    \exp\Big(-\frac{H_{\RealDiso}}{{\BoltCons}T}\Big) \nonumber\\
    &=& \PartitionLiquid(V) \Bigg\ltha \prod _{e=1}^{\LinkNumb}
    \LinkPote\big( \vert \PartPosi_{i_e} - \PartPosi_{j_e} \vert \big)
    \Bigg\rtha_{1}^{\HamiExcl},\end{aligned}$$ where $\HamiExcl \equiv \frac{\ExclVolu }{2}\sum_{i,j=1}^{\PartNumb} \delta (\PartPosi_i-\PartPosi_j)$ is the excluded-volume interaction part of the Hamiltonian, and $\PartitionLiquid(V) \equiv \int _{\Volu} \prod_{i=1}^{\PartNumb} d \PartPosi_i \exp\big(-\HamiExcl/{\BoltCons}T\big)$ is the partition function of the liquid in the absence of any links. The issue of 