Rapid mixing of the flip chain over non-crossing spanning trees

TL;DR

This paper proves that the flip chain for non-crossing spanning trees in convex position rapidly mixes in polynomial time, using Fuss-Catalan structures and comparison techniques.

Contribution

It establishes a polynomial mixing time bound for the flip chain over non-crossing spanning trees, a problem previously lacking such results.

Findings

Mixing time is O(n^8 log n) for the flip chain.

Uses Fuss-Catalan structures to analyze chain behavior.

Provides a comparison argument with Wilson's lattice path chain.

Abstract

We show that the flip chain for non-crossing spanning trees of $n+1$ points in convex position mixes in time $O(n^8\log n)$. We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wilson's lattice path chain (Wilson 2004).