Sampling triangulations of manifolds using Monte Carlo methods

TL;DR

This paper introduces a Monte Carlo approach using biased random walks to efficiently sample and analyze triangulations of manifolds, enabling estimation of rare configurations and growth rates.

Contribution

It presents a novel Monte Carlo method combining Pachner moves with Metropolis-Hastings to sample manifold triangulations uniformly or with bias, applicable to surfaces and 3-manifolds.

Findings

Recovered asymptotic growth rates for sphere triangulations

Observed exponential growth in 3-sphere triangulation types

Provided evidence of fixed edge-degree distributions in large 3-sphere triangulations

Abstract

We propose a Monte Carlo method to efficiently find, count, and sample abstract triangulations of a given manifold M. The method is based on a biased random walk through all possible triangulations of M (in the Pachner graph), constructed by combining (bi-stellar) moves with suitable chosen accept/reject probabilities (Metropolis-Hastings). Asymptotically, the method guarantees that samples of triangulations are drawn at random from a chosen probability. This enables us not only to sample (rare) triangulations of particular interest but also to estimate the (extremely small) probability of obtaining them when isomorphism types of triangulations are sampled uniformly at random. We implement our general method for surface triangulations and 1-vertex triangulations of 3-manifolds. To showcase its usefulness, we present a number of experiments: (a) we recover asymptotic growth rates for the number of isomorphism types of simplicial triangulations of the 2-dimensional sphere; (b) we experimentally observe that the growth rate for the number of isomorphism types of 1-vertex triangulations of the 3-dimensional sphere appears to be singly exponential in the number of their tetrahedra; and (c) we present experimental evidence that a randomly chosen isomorphism type of 1-vertex n-tetrahedra 3-sphere triangulation, for n tending to infinity, almost surely shows a fixed edge-degree distribution which decays exponentially for large degrees, but shows non-monotonic behaviour for small degrees.