Simulating triangulations

Více o knize

Triangulations, which can be understood as a tessellation of space into simplicial building blocks, appear in various branches of physics. They provide a way to describe complex and curved objects in a discretized manner, applicable in foams, gels, and porous media, as well as in fluid simulations and dissipative systems. By interpreting triangulations as (maximal planar) graphs, they can be utilized in graph theory and statistical physics, functioning as small-world networks or networks of spins, and even in biological physics as actin networks. Furthermore, an analogue of the Einstein-Hilbert action on triangulations allows for the formulation of theories of quantum gravity. In mathematics, triangulations play a significant role, especially in discrete topology. Despite their prevalence, fundamental questions remain, such as the number of triangulations for a given set of points or manifold, and whether these can be exponentially many. Another unresolved issue is the ergodicity of elementary steps transforming triangulations in computer simulations. Additionally, when triangulations model spacetime, the existence of a meaningful continuum limit aligning with general relativity is uncertain. This thesis addresses some of these fundamental questions using Markov chain Monte Carlo simulations, a probabilistic method for calculating statistical expectation values and high-dimensional integrals, while also exploring the Wang-Land