A modified c = 1 matrix model with new critical behavior

By introducing a ∫ dt g (TrΦ(t)) 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 ∆ log∆, where ∆ is the cosmological constant (the sum over surfaces of area A goes as A). For comparison, in the conventional c = 1 model the sum over planar surfaces behaves as ∆/ log∆.