Figure A random spanning-tree decorated triangulation of the torus with 100000 triangles viewed via its periodic Tutte embedding in the plane.Figure Another random triangulation with 180000 triangles. Triangles are colored according to their graph distance to a distinguished vertex.Figure Another random triangulation with 250000 triangles.
Video A compilation video of several random triangulations, where we zoom in on the distinguished vertex. This way we get a sense of the fractal nature of two-dimensional quantum gravity.
Video This video also shows the Tutte embedding of a torus triangulation, but this time the triangulation evolves as a Markov chain under uniform triangle flips. This Markov chain is known to approach the uniform random torus triangulation.
Video Similarly to the previous video this shows a triangle-flip Markov chain, but a non-uniform one. It corresponds to a Metropolis-Hastings simulation with Boltzmann weight for each triangulation given by a power of the determinant of the graph Laplacian. When this power, related to the so-called matter central charge, is increased the geometry approaches a semi-classical regimes.
