# 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?