  We present a detailed probabilistic and structural analysis of the set of
weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is
even, to any fixed graph: we show that the corresponding probability
distribution on such homomorphisms is close to a distribution defined
constructively as a certain random perturbation of some dominant phase. This
has several consequences, including solutions (in a strong form) to conjectures
of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include
sharp asymptotics for the number of independent sets and the number of proper
$q$-colourings of $\mathbb{Z}_m^n$ (so in particular, the discrete hypercube).
We give further applications to the study of height functions and (generalised)
rank functions on the discrete hypercube and disprove a conjecture of Kahn and
Lawrenz. For the proof we combine methods from statistical physics, entropy and
graph containers and exploit isoperimetric and algebraic properties of the
torus.
