  Scalar field theory on the fuzzy two-sphere, represented as a hermitian
matrix model that includes kinetic, mass and quartic interaction terms, is
studied. The effective action in the symmetric large-N regime is analyzed using
a self-consistent bootstrap method which fixes its form up to sixth order in
the eigenvalues and gives a closed expression to all orders in the quadratic
invariant of the matrix, valid close to semicircular distributions. Using this
action the eigenvalue distribution is calculated for the interacting theory in
the appropriate scaling limit and the phase transition from the disordered to
the symmetric ordered phase is identified, including nonperturbative effects.
