  This article starts by introducing a new theoretical framework to model
spatial systems which is obtained from the framework of interacting particle
systems by replacing the traditional graphical structure that defines the
network of interactions with a structure of hypergraph. This new perspective is
more appropriate to define stochastic spatial processes in which large blocks
of vertices may flip simultaneously, which is then applied to define a spatial
version of the Galam's majority rule model. In our spatial model, each vertex
of the lattice has one of two possible competing opinions, say opinion 0 and
opinion 1, as in the popular voter model. Hyperedges are updated at rate one,
which results in all the vertices in the hyperedge changing simultaneously
their opinion to the majority opinion of the hyperedge. In the case of a tie in
hyperedges with even size, a bias is introduced in favor of type 1, which is
motivated by the principle of social inertia. Our analytical results along with
simulations and heuristic arguments suggest that, in any spatial dimensions and
when the set of hyperedges consists of the collection of all n x ... x n blocks
of the lattice, opinion 1 wins when n is even while the system clusters when n
is odd, which contrasts with results about the voter model in high dimensions
for which opinions coexist. This is fully proved in one dimension while the
rest of our analysis focuses on the cases when n = 2 and n = 3 in two
dimensions.
