  We present density split statistics, a framework that studies lensing and
counts-in-cells as a function of foreground galaxy density, thereby providing a
large-scale measurement of both 2-point and 3-point statistics. Our method
extends our earlier work on trough lensing and is summarized as follows: given
a foreground (low redshift) population of galaxies, we divide the sky into
subareas of equal size but distinct galaxy density. We then measure lensing
around uniformly spaced points separately in each of these subareas, as well as
counts-in-cells statistics (CiC). The lensing signals trace the matter density
contrast around regions of fixed galaxy density. Through the CiC measurements
this can be related to the density profile around regions of fixed matter
density. Together, these measurements constitute a powerful probe of cosmology,
the skewness of the density field and the connection of galaxies and matter.
  In this paper we show how to model both the density split lensing signal and
CiC from basic ingredients: a non-linear power spectrum, clustering hierarchy
coefficients from perturbation theory and a parametric model for galaxy bias
and shot-noise. Using N-body simulations, we demonstrate that this model is
sufficiently accurate for a cosmological analysis on year 1 data from the Dark
Energy Survey.
