Summary: How quickly does the edge-flip random walk in the space of triangulations of a closed, connected, orientable surface converge to the uniform distribution over all triangulations?
Let $M$ be a closed, connected, orientable 2-manifold. Given a triangulation $T$ with no self loops and an edge $e \in T$ s.t. the triangles adjacent to $e$ shared only two vertices, we can define a new triangulation $T'$ by removing $e$ and retriangulating the resulting quad the other way. This operation is called an edge flip.
Let $\mathcal{T_n}$ be the set of triangulations of $M$ with $n$ vertices and no self loops, and make $\mathcal{T_n}$ into an unordered graph where the graph edges are edge flips. I believe, but do not have a proof or a reference, that $\mathcal{T_n}$ is a connected graph. We can define a random walk on $\mathcal{T_n}$ by choosing either not to flip or any safe edge flip uniformly at random. If the graph is connected, this random walk eventually converges to the uniform distribution. How long does this take?
This paper provides bounds on this problem for the case of a convex polygon with no internal vertices, but a brief search didn't find any results for the general case: