  In the recent years, field theory on non-commutative (NC) spaces has
attracted a lot of attention. Most literature on this subject deals with
perturbation theory, although the latter runs into grave problems beyond one
loop. Here we present results from a fully non-perturbative approach. In
particular, we performed numerical simulations of the \lambda \phi^{4} model
with two NC spatial coordinates, and a commutative Euclidean time. This theory
is lattice discretized and then mapped onto a matrix model. The simulation
results reveal a phase diagram with various types of ordered phases. We discuss
the suitable order parameters, as well as the spatial and temporal correlators.
The dispersion relation clearly shows a trend towards the expected IR
singularity. Its parameterization provides the tool to extract the continuum
limit.
