Section \[SEC:FTMF\], we address the interactions in Eq. (\[EQ:HRLPM\]) by eliminating the particle coordinates in favor of collective fields, which have the form of joint densities of the replicas of the particles. This continuum approach enables us to focus on the physics of random particle localization, particularly at lengthscales that are relevant for such localization, which (except when the density of links is extremely high) are long compared with the ranges of $\VExc$ and $\VLink$. With a focus on these longer lengthscales in mind, we see that it is adequate to replace the repulsive interaction $\VExc(\PartPosi)$ by the model Dirac delta-function excluded-volume interaction $\ExclVolu\,\delta(\PartPosi)$, characterized by the strength $\ExclVolu$ [@Gennes1979; @Deam1976; @Doi1986]. This procedure amounts to making a gradient expansion in real space (or, equivalently, a wave vector expansion in Fourier space) of $\VExc$ and retaining only the zeroth-order term; it gives for the strength $\ExclVolu$ the value ${\int}d\PartPosi\,\VExc(\PartPosi)$. Terms of higher order in the gradient expansion would have a non-negligible impact on the suppression of density fluctuations only at lengthscales comparable to or shorter than the range of $\VExc$, and fluctuation modes at such lengthscales are not the ones driven via random linking to the instability associated with random 