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.

翻译：我们提出了一种蒙特卡洛方法，用于高效地寻找、计数和采样给定流形M的抽象三角剖分。该方法基于一条通过M所有可能三角剖分（在帕赫纳图(Pachner graph)中）的偏置随机游走，通过将（双星形）移动与适当选择的接受/拒绝概率（梅特罗波利斯-黑斯廷斯算法）相结合来构建。渐近地，该方法保证从选定概率分布中随机抽取三角剖分样本。这不仅使我们能够采样特别感兴趣的（罕见）三角剖分，还能估计当均匀随机抽取三角剖分的同构类型时获得它们的（极小的）概率。我们将该通用方法应用于曲面三角剖分和三维流形的单顶点三角剖分。为展示其实用性，我们进行了一系列实验：(a) 恢复了二维球面单纯三角剖分同构类型数量的渐近增长率；(b) 实验观察到，三维球面单顶点三角剖分同构类型数量的增长率似乎随四面体数量呈单指数增长；(c) 实验证据表明，对于趋于无穷大的n，随机选取的单顶点n四面体三维球面三角剖分同构类型几乎必然呈现出固定的边度分布，该分布在大度时呈指数衰减，但在小度时表现出非单调行为。