### Abstract

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

Pages (from-to) | 469–521 |

Journal | Probability Theory and Related Fields |

Volume | 169 |

Issue number | 1-2 |

Early online date | 10 Aug 2016 |

DOIs | |

Publication status | Published - 1 Oct 2017 |

### Fingerprint

### Keywords

- math.PR
- cs.DM
- math-ph
- math.CO
- math.MP

### Cite this

**A Lyapunov function for Glauber dynamics on lattice triangulations.** / Stauffer, Alexandre.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 169, no. 1-2, pp. 469–521. https://doi.org/10.1007/s00440-016-0735-z

}

TY - JOUR

T1 - A Lyapunov function for Glauber dynamics on lattice triangulations

AU - Stauffer, Alexandre

PY - 2017/10/1

Y1 - 2017/10/1

N2 - We study random triangulations of the integer points [0,n]2∩Z2 , where each triangulation σ has probability measure λ|σ| with λ>0 being a real parameter and |σ| denoting the sum of the length of the edges in σ . Such triangulations are called lattice triangulations. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime λ<1 , the function behaves as a Lyapunov function with respect to Glauber dynamics; that is, the function is a supermartingale. We show the applicability of the above result by establishing several features of lattice triangulations, such as tightness of local measures, exponential tail of edge lengths, crossings of small triangles, and decay of correlations in thin rectangles. These are the first results on lattice triangulations that are valid in the whole subcritical regime λ<1 . In a very recent work with Caputo, Martinelli and Sinclair, we apply this Lyapunov function to establish tight bounds on the mixing time of Glauber dynamics in thin rectangles that hold for all λ<1 . The Lyapunov function result here holds in great generality; it holds for triangulations of general lattice polygons (instead of the [0,n]2 square) and also in the presence of arbitrary constraint edges.

AB - We study random triangulations of the integer points [0,n]2∩Z2 , where each triangulation σ has probability measure λ|σ| with λ>0 being a real parameter and |σ| denoting the sum of the length of the edges in σ . Such triangulations are called lattice triangulations. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime λ<1 , the function behaves as a Lyapunov function with respect to Glauber dynamics; that is, the function is a supermartingale. We show the applicability of the above result by establishing several features of lattice triangulations, such as tightness of local measures, exponential tail of edge lengths, crossings of small triangles, and decay of correlations in thin rectangles. These are the first results on lattice triangulations that are valid in the whole subcritical regime λ<1 . In a very recent work with Caputo, Martinelli and Sinclair, we apply this Lyapunov function to establish tight bounds on the mixing time of Glauber dynamics in thin rectangles that hold for all λ<1 . The Lyapunov function result here holds in great generality; it holds for triangulations of general lattice polygons (instead of the [0,n]2 square) and also in the presence of arbitrary constraint edges.

KW - math.PR

KW - cs.DM

KW - math-ph

KW - math.CO

KW - math.MP

UR - http://dx.doi.org/10.1007/s00440-016-0735-z

UR - http://dx.doi.org/10.1007/s00440-016-0735-z

U2 - 10.1007/s00440-016-0735-z

DO - 10.1007/s00440-016-0735-z

M3 - Article

VL - 169

SP - 469

EP - 521

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1-2

ER -