A modified c=1 matrix model with new critical behavior

By introducing a $int dt , gleft(Tr Phi^2(t)right)^2$ term into the action of the $c=1$ matrix model of two-dimensional quantum gravity, we find a new critical behavior for random surfaces. The planar limit of the path integral generates multiple spherical ``bubbles which touch one another at single points. At a special value of $g$, the sum over connected surfaces behaves as $Delta^2 logDelta$, where $Delta$ is the cosmological constant (the sum over surfaces of area $A$ goes as $A^{-3}$). For comparison, in the conventional $c=1$ model the sum over planar surfaces behaves as $Delta^2/ logDelta$.