### Abstract

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2
^{-s} , (a + 1)2
^{-s} ] × [b2
^{-t} , (b + 1)2
^{-t} ] for a, b, s, t EZ≥ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n
^{4.09} ), which implies that the mixing time is at most O(n
^{5.09} ). We complement this by showing that the relaxation time is at least Ω(n
^{1.38} ), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

Original language | English |
---|---|

Pages (from-to) | 365-387 |

Number of pages | 23 |

Journal | Combinatorics, Probability and Computing |

Volume | 28 |

Issue number | 3 |

Early online date | 31 Oct 2018 |

DOIs | |

Publication status | Published - 1 May 2019 |

### Fingerprint

### Keywords

- Random dyadic tilings
- Rapid mixing
- Spectral gap

### ASJC Scopus subject areas

- Theoretical Computer Science
- Applied Mathematics
- Statistics and Probability
- Computational Theory and Mathematics

### Cite this

*Combinatorics, Probability and Computing*,

*28*(3), 365-387. https://doi.org/10.1017/S0963548318000470

**Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings.** / Cannon, Sarah; Levin, David A.; Stauffer, Alexandre.

Research output: Contribution to journal › Article

*Combinatorics, Probability and Computing*, vol. 28, no. 3, pp. 365-387. https://doi.org/10.1017/S0963548318000470

}

TY - JOUR

T1 - Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings

AU - Cannon, Sarah

AU - Levin, David A.

AU - Stauffer, Alexandre

PY - 2019/5/1

Y1 - 2019/5/1

N2 - We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2 -s , (a + 1)2 -s ] × [b2 -t , (b + 1)2 -t ] for a, b, s, t EZ≥ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n 4.09 ), which implies that the mixing time is at most O(n 5.09 ). We complement this by showing that the relaxation time is at least Ω(n 1.38 ), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

AB - We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2 -s , (a + 1)2 -s ] × [b2 -t , (b + 1)2 -t ] for a, b, s, t EZ≥ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n 4.09 ), which implies that the mixing time is at most O(n 5.09 ). We complement this by showing that the relaxation time is at least Ω(n 1.38 ), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

KW - Random dyadic tilings

KW - Rapid mixing

KW - Spectral gap

UR - http://www.scopus.com/inward/record.url?scp=85056140355&partnerID=8YFLogxK

U2 - 10.1017/S0963548318000470

DO - 10.1017/S0963548318000470

M3 - Article

VL - 28

SP - 365

EP - 387

JO - Combinatorics, Probability and Computing

JF - Combinatorics, Probability and Computing

SN - 0963-5483

IS - 3

ER -