Improved mixing for the convex polygon triangulation flip walk
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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.
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