Crumpled Triangulations and Critical Points in 4D Simplicial Quantum Gravity

We estimate analytically the critical coupling separating the weak and the strong coupling regime in 4D simplicial quantum gravity to be located at k~ tit '~ 1.3093. By carrying out a detailed geometrical analysis of the strong coupling phase we argue that the distribution of dynamical triangulations with singular vertices and singular edges, dominating in such a regime, is characterized by distinct subdominating peaks. The presence of such peaks generates volumedependent pseudo-critical points: kocrit(N4 = 32000) ~ 1.25795, k~rit(N4 = 48000) --~ 1.26752, k~rit(N4 = 64000) '~ 1.27466, etc., which appear in good agreement with available Monte Carlo data. Under a certain scaling hypothesis we analytically characterize the (canonical) average value, cl (N4; k2) = (No)/N4, and the susceptibility, c2(N4; k2) = ((N 2) - (No)2)/N4, associated with the vertex distribution of the 4D triangulations considered. Again, the resulting analytical expressions are found in quite a good agreement with their Monte Carlo counterparts