Markov processes in the dynamics of primitive triangulations in spaces R3 and R4

Abstract

Lattice models and simplicial complexes continue to play an important role in theoretical physics and gain an increasing interest in connection with the application of dynamic triangulations to the construction of quantum gravity models. With the advent of modern supercomputers, the piecewise-linear complexes and the bistellar transformations become a basis of numerical methods in combinatorial geometry and topology. In this paper, random flips of primitive triangulations in space R³ with vertices from an integer set Z³ are considered as Markov chains and their properties of periodicity, decomposability, and ergodicity are studied. As a result, an asymptotic behavior of the triangulated space as a whole is determined. Similar methods are proposed for primitive triangulations in space R⁴.