  We present a first numerical investigation of a non-commutative gauge theory
defined via the spectral action for Moyal space with harmonic propagation. This
action is approximated by finite matrices. Using Monte Carlo simulation we
study various quantities such as the energy density, the specific heat density
and some order parameters, varying the matrix size and the independent
parameters of the model. We find a peak structure in the specific heat which
might indicate possible phase transitions. However, there are mathematical
arguments which show that the limit of infinite matrices is very different from
the original spectral model.
