  We provide a fully analytical microscopic theory for the proton correlations
in water ice $I_h$. We compute the full diffuse elastic neutron scattering
structure factor, which we find to be in excellent quantitative agreement with
Monte Carlo simulations. It is also in remarkable qualitative agreement with
experiment, in the absence of any fitting parameters. Our theory thus provides
a tractable analytical starting point to account for more delicate features of
the proton correlations in water ice. In addition, it directly determines an
effective field theory of water ice as a topological phase.

Statistical Mechanics;Strongly Correlated Electrons