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

Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, Mark Jerrum, Jiaheng Wang

TL;DR

The paper addresses rapid mixing for the flip chain on non-crossing spanning trees (NCSTs) of $n+1$ points in convex position. It develops a Markov-chain comparison framework by coupling the NCST flip chain to a simpler adjacent-move chain on $2$-Dyck paths, leveraging Fuss–Catalan structures and a Wilson-style coupling to obtain an $O(n^3 \log n)$ mixing bound for the simpler chain. By carefully simulating each adjacent move with a controlled sequence of NCST edge flips and a shifting operation, the authors bound the path congestion and transfer the mixing time to the NCST flip chain, yielding an overall $O(n^8 \log n)$ time bound. This establishes the first polynomial upper bound for NCST flip-chain mixing in convex position and highlights deep connections between NCSTs, Fuss-Catalan combinatorics, and lattice-path Markov chains, with potential implications for approximate counting and reconfiguration problems.

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).

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

TL;DR

The paper addresses rapid mixing for the flip chain on non-crossing spanning trees (NCSTs) of points in convex position. It develops a Markov-chain comparison framework by coupling the NCST flip chain to a simpler adjacent-move chain on -Dyck paths, leveraging Fuss–Catalan structures and a Wilson-style coupling to obtain an mixing bound for the simpler chain. By carefully simulating each adjacent move with a controlled sequence of NCST edge flips and a shifting operation, the authors bound the path congestion and transfer the mixing time to the NCST flip chain, yielding an overall time bound. This establishes the first polynomial upper bound for NCST flip-chain mixing in convex position and highlights deep connections between NCSTs, Fuss-Catalan combinatorics, and lattice-path Markov chains, with potential implications for approximate counting and reconfiguration problems.

Abstract

We show that the flip chain for non-crossing spanning trees of points in convex position mixes in time . We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wilson's lattice path chain (Wilson 2004).
Paper Structure (14 sections, 11 theorems, 27 equations, 13 figures, 1 algorithm)

This paper contains 14 sections, 11 theorems, 27 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Technical overview
  3. Related work
  4. Open problems
  5. Markov chain comparison
  6. Path method
  7. Fuss-Catalan numbers and structures
  8. Dyck paths
  9. Non-crossing spanning trees
  10. Spectral gap of the adjacent move chain
  11. Adjacent move versus flip move
  12. Characterise the adjacent move via flip moves
  13. Path construction
  14. Bound the congestion

Key Result

Theorem 1

For $n+1$ points on the plane in convex position, the mixing time of the flip chain over their non-crossing spanning trees is $O(n^8\log n)$.

Figures (13)

  • Figure 1: Illustration of a flip move. The red, dashed edge in the leftmost figure is dropped. This gives $9$ possible edges (dotted in the middle figure) that can be added back to form a valid NCST. The blue, thick edge in the rightmost figure is picked.
  • Figure 2: NCSTs corresponding to the $2$-Dyck paths in \ref{['exp:bad-correspondence']}.
  • Figure 3: Decomposing \ref{['emp:dyck-path']} by \ref{['fac:dyck-path-decompose']}.
  • Figure 6: An example of a shift sequence. The black longer bubbles represent minimal segments, and there may be $0$ or more of them. The red shorter bubbles represent valid sub-trees and are not necessarily minimal.
  • Figure : (a)
  • ...and 8 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Example 2
  • Theorem 3: Theorem 2.1 of DS93
  • Remark
  • Definition 4: $k$-Dyck path
  • Example 5
  • Corollary 8
  • Proposition 9
  • proof
  • Definition 10
  • ...and 16 more