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The friends-of-friends algorithm (hereafter, FOF) is a percolation algorithmwhich is routinely used to identify dark matter halos from N-body simulations.We use results from percolation theory to show that the boundary of FOF halosdoes not correspond to a single density threshold but to a range of densitiesclose to a critical value that depends upon the linking length parameter, b. Weshow that for the commonly used choice of b = 0.2, this critical density isequal to 81.62 times the mean matter density. Consequently, halos identified bythe FOF algorithm enclose an average overdensity which depends on their densityprofile (concentration) and therefore changes with halo mass contrary to thepopular belief that the average overdensity is ~180. We derive an analyticalexpression for the overdensity as a function of the linking length parameter band the concentration of the halo. Results of tests carried out using simulatedand actual FOF halos identified in cosmological simulations show excellentagreement with our analytical prediction. We also find that the mass of thehalo that the FOF algorithm selects crucially depends upon mass resolution. Wefind a percolation theory motivated formula that is able to accurately correctfor the dependence on number of particles for the mock realizations ofspherical and triaxial Navarro-Frenk-White halos. However, we show that thiscorrection breaks down when applied to the real cosmological FOF halos due topresence of substructures. Given that abundance of substructure depends onredshift and cosmology, we expect that the resolution effects due tosubstructure on the FOF mass and halo mass function will also depend onredshift and cosmology and will be difficult to correct for in general.Finally, we discuss the implications of our results for the universality of themass function.

The overdensity and masses of the friends-of-friends halos and universality of the halo mass function