  Double Hurwitz numbers count covers of the projective line by genus g curves
with assigned ramification profiles over 0 and infinity, and simple
ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown
double Hurwitz numbers are piecewise polynomial in the orders of ramification,
and Shadrin, Shapiro and Vainshtein have determined the chamber structure and
wall crossing formulas for g=0. This paper gives a unified approach to these
results and strengthens them in several ways -- the most important being the
extension of the results of Shapiro, Shadrin and Vainshtein to arbitrary genus.
The main tool is the authors' previous work expressing double Hurwitz number as
a sum over certain labeled graphs. We identify the labels of the graphs with
lattice points in the chambers of certain hyperplane arrangements, which are
well known to give rise to piecewise polynomial functions. Our understanding of
the wall crossing for these functions builds on the work of Varchenko, and
could have broader applications.
