Improved mixing for the convex polygon triangulation flip walk

We prove that the well-studied triangulation flip walk on a convex point set mixes in time $O(n^{4.75}),$ the first progress since McShine and Tetali's $O(n^5 \log n)$ bound in 1997. In the process we determine the expansion of the associahedron graph $K_n$ up to a factor of $O(n^{3/4})$. To obtain these results, we extend a framework we developed in a previous preprint--extending the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda--for establishing conditions under which the Glauber dynamics on independent sets and other combinatorial structures mix rapidly.