Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Block coupling and rapidly mixing k-heights

Stefan Felsner, Daniel Heldt, Sandro Roch, Peter Winkler

Abstract

A $k$-height on a graph $G=(V, E)$ is an assignment $V\to\{0, \ldots, k\}$ such that the value on ajacent vertices differs by at most $1$. We study the Markov chain on $k$-heights that in each step selects a vertex at random, and, if admissible, increases or decreases the value at this vertex by one. In the cases of $2$-heights and $3$-heights we show that this Markov chain is rapidly mixing on certain families of grid-like graphs and on planar cubic $3$-connected graphs. The result is based on a novel technique called block coupling, which is derived from the well-established monotone coupling approach. This technique may also be effective when analyzing other Markov chains that operate on configurations of spin systems that form a distributive lattice. It is therefore of independent interest.

Block coupling and rapidly mixing k-heights

Abstract

A -height on a graph is an assignment such that the value on ajacent vertices differs by at most . We study the Markov chain on -heights that in each step selects a vertex at random, and, if admissible, increases or decreases the value at this vertex by one. In the cases of -heights and -heights we show that this Markov chain is rapidly mixing on certain families of grid-like graphs and on planar cubic -connected graphs. The result is based on a novel technique called block coupling, which is derived from the well-established monotone coupling approach. This technique may also be effective when analyzing other Markov chains that operate on configurations of spin systems that form a distributive lattice. It is therefore of independent interest.

Paper Structure

This paper contains 19 sections, 21 theorems, 86 equations, 8 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Obtaining k-heights from alpha-orientations
  3. Rapidly mixing Markov chains on k-heights
  4. Outline
  5. Preliminaries
  6. Markov chains, mixing time and couplings
  7. The up/down Markov chain
  8. The block Markov chain
  9. A discrete version of a theorem by Strassen and the block divergence
  10. Mixing time of up/down and block Markov chain
  11. Stochastic dominance in block Markov chain
  12. Block coupling
  13. Applications
  14. k-heights on toroidal rectangular grid graphs
  15. k-heights on toroidal hexagonal grid graphs
  16. ...and 4 more sections

Key Result

Theorem 1

Let $G=(V, E)$ be a finite graph, and $\mathcal{B}$ be a finite family of blocks, such that for every vertex $v\in V$ there is at least one $B\in\mathcal{B}$ with $v\in B$. If there exists $\beta < 1$ such that for all $v \in V$ then for the mixing time $\tau(\varepsilon)$ of the up/down Markov chain $\mathcal{M}$ on $k$-heights of $G$ we have where with $m := \max_{v\in V} \#\{ B\in\mathcal{B}

Figures (8)

  • Figure 1: Example of a $2$-height. Colors visualize the values as a heat map.
  • Figure 2: (a) Example of an $\alpha$-orientation $\overrightarrow{E}_0$. (b) the $\alpha$-orientation obtained by flipping the face bounded by the red arcs. (c) Minimal $\alpha$-orientation~$\overrightarrow{E}_{\text{min}}$. Red numbers record the number of face flips needed to reach $\overrightarrow{E}_0$.
  • Figure 3: Toroidal rectangular grid graph of size $12\times 9$.
  • Figure 4: Hexagonal grid (a) or dual (b) of toroidal triangle grid of size $10\times 8$. Red vertices or faces form a block; blue vertices or faces are part of the boundary $\partial B$.
  • Figure 5: Example of a dual graph (black, solid) of a $4$-connected triangulation (orange, dashed)
  • ...and 3 more figures

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 1
  • Theorem 2: Dyer & Greenhill, Theorem 2.1 in dyerGreenhill
  • Theorem 3: Dyer & Greenhill, Theorem 2.2 in dyerGreenhill
  • Theorem 4: Randall & Tetali, Theorem 3 in randallTetali00
  • Lemma 1
  • proof
  • Theorem 5
  • Lemma 2
  • proof
  • ...and 24 more