  Motivated by understanding the phase structure of $d >1$ strings we
investigate the $c=1$ matrix model with $g' (\tr M(t)^{2})^{2}$ interaction
which is the simplest approximation of the model expected to describe the
critical phenomena of the large-$N$ reduced model of odd-dimensional matrix
field theory. We find three distinct phases: (i) an ordinary $c=1$ gravity
phase, (ii) a branched polymer phase and (iii) an intermediate phase. Further
we can also analyse the one with slightly generalized $ g^{(2)} (\frac{1}{N}\tr
M(t)^{2})^{2} +g^{(3)} (\frac{1}{N}\tr M(t)^{2})^{3} + \cdots + g^{(n)}
(\frac{1}{N}\tr M(t)^{2})^{n} $ interaction. As a result the multi-critical
versions of the phase (ii) are found.
