  We use self-similarity in N-body simulations of scale-free models to test for
resolution dependence in the mass function and two-point correlation functions
of dark matter halos. We use 1024$^3$ particle simulations performed with
ABACUS, and compare results obtained with two halo finders: friends-of-friends
(FOF) and ROCKSTAR. The FOF mass functions show a systematic deviation from
self-similarity which is explained by resolution dependence of the FOF mass
assignment previously reported in the literature. Weak evidence for convergence
is observed only starting from halos of several thousand particles, and mass
functions are overestimated by at least as much as 20-25 percent for halos of
50 particles. The mass function of the default ROCKSTAR halo catalog (with
bound virial spherical overdensity mass), on the other hand, shows good
convergence from of order 50 to 100 particles per halo, with no detectable
evidence at the few percent level of any systematic dependence for larger
particle number. Tests show that the mass unbinding procedure in ROCKSTAR is
the key factor in obtaining this much improved resolution. Applying the same
analysis to the halo-halo two point correlation function, we find again strong
evidence for convergence only for ROCKSTAR halos, at separations sufficiently
large so that halos do not overlap. At these separations we can exclude
dependence on resolution at the 5-10 percent level once halos have of order 50
to 100 particles. At smaller separations results are not converged even at
significantly larger particle number, and bigger simulations would be required
to establish the resolution required for convergence.
