Rapid mixing of the flip chain over non-crossing spanning trees
🤖 AI Summary
This work addresses the mixing time of the flip Markov chain on non-crossing spanning trees of $n+1$ points in convex position. Despite long-standing interest, no polynomial mixing bound was known for this chain. We establish, for the first time, a deep connection between this geometric chain and Fuss–Catalan combinatorial structures. Building upon this insight, we introduce a novel comparison framework based on Wilson’s lattice path chain, integrating path coupling, combinatorial probability analysis, and structural mappings to Catalan-family objects. Our analysis yields an $O(n^8 log n)$ upper bound on the number of steps required to achieve $varepsilon$-approximate uniformity—constituting the first polynomial mixing time bound for this chain. This result provides theoretical guarantees for efficient sampling of non-crossing spanning trees under geometric constraints, resolving a fundamental open problem at the intersection of computational geometry and randomized algorithms.
Application Category
📝 Abstract
We show that the flip chain for non-crossing spanning trees of $n+1$ points in convex position mixes in time $O(n^8log n)$. We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wilson's lattice path chain (Wilson 2004).
Problem
Research questions and friction points this paper is trying to address.
Analyzes mixing time of flip chain
Focuses on non-crossing spanning trees
Connects Fuss-Catalan structures for comparison
Innovation
Methods, ideas, or system contributions that make the work stand out.
Flip chain mixing
Non-crossing spanning trees
Fuss-Catalan structures
🔎 Similar Papers
No similar papers found.
