Peak-Background Split, Renormalization, and Galaxy Clustering
  We present a derivation of two-point correlations of general tracers in the
peak-background split (PBS) framework by way of a rigorous definition of the
PBS argument. Our expressions only depend on connected matter correlators and
"renormalized" bias parameters with clear physical interpretation, and are
independent of any coarse-graining scale. This result should be contrasted with
the naive expression derived from a local bias expansion of the tracer number
density with respect to the matter density perturbation \delta_L coarse-grained
on a scale R_L. In the latter case, the predicted tracer correlation function
receives contributions of order <\delta_L^n> at each perturbative order n,
whereas, in our formalism, these are absorbed in the PBS bias parameters at all
orders. Further, this approach naturally predicts both a scale-dependent bias ~
k^2 such as found for peaks of the density field, and the scale-dependent bias
induced by primordial non-Gaussianity in the initial conditions. The only
assumption made about the tracers is that their abundance at a given position
depends solely on the matter distribution within a finite region around that
position.
