Why is the boundary of friends-of-friends (FOF) halo corresponding to iso-density contour? The friends-of-friends algorithm (hereafter FOF) is commonly used to find halos in cosmological simulations. (For more information, please refer to here and here)
I found that some literature argues that the boundary of the FOF halo is corresponding to an iso-density surface. For example, Springel et al 2001 states

It
places any two particles with a separation less than some linking
length $b$ into the same group. In this way, particle groups are
formed that correspond to regions approximately enclosed by
isodensity contours with threshold value $\rho\propto1/b^3$.

More et al 2011 says

One could expect that for a given value of $b$, the FOF algorithm defines the boundary of a halo as corresponding to a certain isodensity surface, at least in the limit of large number of particles.

However, I can't understand the connection between the boundary of the FOF group of particles and iso-density contour. How to derive this connection in formula form? Does this connection need other hypotheses?
If you have enough particles to sample your field, then a given (local) density corresponds to the same mean distance between your particles. The FOF algorithm keeps linking particles to a halo until a particle is farther away than the linking length. That means that the halo boundary will be ~exactly on the border between regions of interparticle distances smaller and larger than the linking length, corresponding to a boundary between larger and smaller densities, respectively. That is, an iso-density surface.
In contrast, a halo finder using the spherical overdensity (SO) algorithm will always result in a spherical boundary, which will only be an iso-density surface in the case of a completely symmetrical halo.
This is the reason that FOF halo finders tend to include more filaments outside the region identified by SO halo finder. This may or may not be desired.