  The fuzzy disc is a discretization of the algebra of functions on the two
dimensional disc using finite matrices which preserves the action of the
rotation group. We define a $\varphi^4$ scalar field theory on it and analyze
numerically for three different limits for the rank of the matrix going to
infinity. The numerical simulations reveal three different phases: uniform and
disordered phases already the present in the commutative scalar field theory
and a nonuniform ordered phase as a noncommutative effects. We have computed
the transition curves between phases and their scaling. This is in agreement
with studies on the fuzzy sphere, although the speed of convergence for the
disc seems to be better. We have performed also three the limits for the theory
in the cases of the theory going to the commutative plane or commutative disc.
In this case the theory behaves differently, showing the intimate relationship
between the nonuniform phase and noncommutative geometry.
