Analytical vs. Numerical Data

Abstract

The interplay between the asymptotic estimates arising from (6.102) and the results coming from Monte Carlo simulations requires some care, since there are manifolds which are algorithmically unrecognizable. Denote such a manifold by M₀, assume that n = 4, and let M₀ be finitely described by a dynamical triangulation T(M₀). Monte Carlo simulations make use of two kinds of moves altering the initial triangulation T(M₀): a finite set of local moves [2, 96, 97, 67], and global baby universe surgery moves. The local moves are ergodic, since given any two distinct triangulations of the manifold it is possible in a finite number of moves, carried out successively, to change on triangulation into the other. The algorithmic unrecognizability of M₀ means that there exists no algorithm which allows us to decide wether another manifold M, again finitely described by a triangulation T(M), is combinatorially equivalent (in the PL-sense), to M₀. It has been proved[95] that the number of ergodic moves[13] needed to connect two triangulations of M₀, T and T*, with N₄(T) = N₄(T*), cannot be bounded by any recursive function of N₄. This result implies that there can be very large barriers between some classes of triangulations of M₀ and that there would be triangulations which can never be reached in any reasonable number of steps, to the effect that the local moves, although ergodic, can be computationally non-ergodic. The manifold used in actual Monte Carlo simulations of 4-dimensional simplicial gravity is mainly the four- sphere $\mathbb{S}^4$,(see [26] for other topologies), and it is not presently known if it is algorithmically recognizable, and no large barriers have been observed yet [11].