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?