Combinatorial and topological phase structure of non-perturbative n-dimensional quantum gravity

W e provide a non-perturbative geom etrical characterization of the partition function of n -dim ensional quantum gravity based on a coarse classi(cid:12)cation of riem annian geom etries. W e show that,under naturalgeom etricalconstraints,the theory adm itsa continuum lim itwith a non-trivialphasestructureparam etrized by the hom otopy types of the class of m anifolds considered. The results obtained qualitatively coincide, when specialized to dim ension two, with those of two-dim ensionalquantum gravity m odelsbased on random triangulations ofsurfaces.