of the RLPM in detail.

Field-theoretic description of the RLPM {#SEC:FTMF}
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We shall use field-theoretic methods to analyze the disorder-averaged free energy $\lda\HelmFreeEner\rda$ and, more specifically, the replicated partition function $Z_{1+n}$. To do this, we introduce a joint probability distribution for the particle density in the replicated space, i.e., the replicated density function $$\begin{aligned}
\label{EQ:DensReal}
    \DensFunc(\REPX)\equiv \frac{1}{\PartNumb} \sum_{i=0}^{\PartNumb}
        \ReplProd \delta^{(d)}(x\REPa-\PartPosi_i\REPa) ,\end{aligned}$$ where $\REPX \equiv (x^{0},x^{1},\ldots,x^{n})$ is a short-hand for the $(1+n)$-replicated position $\Dime$-vector. For convenience, we introduce a complete orthonormal basis set in replica space $\{\BASE\REPa\}_{\alpha=0}^{n}$, in terms of which a vector $\REPX$ can be expressed as $$\begin{aligned}
    \REPX=\ReplSum x\REPa \BASE\REPa .\end{aligned}$$ Note that the components $x\REPa$ are themselves $\Dime$-vectors. With this notation, the density function of a single replica $\alpha$ is given by $\DensFunc_{p\REPa\BASE\REPa}$, which is the Fourier transform of the $\DensFunc(\REPX)$ field, $\DensFunc_{\REPP}$, with momentum nonzero only in replica $\alpha$, corresponding to integrating over the normalized densities in other replicas in real space.

The replicated partition function (\[EQ:ReplPart\]) can be written as a functional of the replicated density function $\DensFunc$ in momentum space as $$\begin{aligned}
\label{EQ:ZQ}
    Z_{1+n} = \int_{\Volu_0} \prod_{i=1}^{\PartNumb} d\PartPosi_{i}^{0}
                        \int_{\Volu} \ReplProd \prod_{j=1}^{\PartNumb} d\PartPosi_{j}\REPa
                        \, e^{-\frac{H_{\DensFunc}\lbrack \DensFunc_{\REPP}\rbrack}{\BoltCons T}} ,\end{aligned}$$ with $$\begin{aligned}
\label{EQ:HQ}
    H_{\DensFunc}\lbrack \DensFunc_{\REPP}\rbrack
    &\equiv& -\frac{\PartNumb 