Systematic errors in weighted 2-point correlation functions: An
  application to interaction-induced star formation
  Weighted correlation functions are an increasingly important tool for
understanding how galaxy properties depend on their separation from each other.
We use a mock galaxy sample drawn from the Millenium simulation, assigning
weights using a simple prescription to illustrate and explore how well a
weighted correlation function recovers the true radial dependence of the input
weights. We find that the use of a weighted correlation function results in a
dilution of the magnitude of any radial dependence of properties and a smearing
out of that radial dependence in radius, compared to the input behavior. We
present a quantitative discussion of the dilution in the magnitude of radial
dependence in properties in the special case of a constant enhancement at r <
rc. In this particular case where there was a SFR enhancement at small radius r
< rc = 35 kpc, the matching of one member of an enhanced pair with a
non-enhanced galaxy in the same group gives an artificial enhancement out to
large radius ~ 200 kpc. We compare this with observations of SFR enhancement
from the SDSS (Li et al. 2008; MNRAS, 385, 1903) finding very similar behavior
- a significant enhancement at radii < 40 kpc and a weak enhancement out to
more than 150 kpc. While we explore a particular case in this Letter, it is
easy to see that the phenomenon is general, and precision analyses of weighted
correlation functions will need to account carefully for this effect using
simulated mock catalogs.
