  This article studies discrete height functions on the discrete hypertorus.
These are functions on the vertices of this hypertorus graph for which the
derivative satisfies a specific condition on each edge. We then perform a
random walk on the set of such height functions, in the spirit of Diffusivity
of a random walk on random walks, a work of Boissard, Cohen, Espinasse, and
Norris. The goal is to estimate the diffusivity of this random walk in the mesh
limit. It turns out that each height functions is characterised by a number of
so-called fractures of the hypertorus. These fractures are then studied in
isolation; we are able to understand their asymptotic behaviour in the mesh
limit due to the recent understanding of the associated random surfaces. This
allows for an asymptotic reduction to a one-dimensional continuous system
consisting of $\operatorname{gcd}\mathbf n$ parts where $\mathbf n\in\mathbb
N^d$ is the fundamental parameter of the original model. We then prove that the
diffusivity of the random walk tends to $1/(1+2\operatorname{gcd}\mathbf n)$ in
this mesh limit.

Probability