A Lyapunov function for Glauber dynamics on lattice triangulations

Research output: Contribution to journalArticle

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Abstract

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.
Original languageEnglish
Pages (from-to)469–521
JournalProbability Theory and Related Fields
Volume169
Issue number1-2
Early online date10 Aug 2016
DOIs
Publication statusPublished - 1 Oct 2017

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Glauber Dynamics
Lyapunov Function
Triangulation
Rectangle
Supermartingale
Mixing Time
Decay of Correlations
Integer Points
Lyapunov function
Tightness
Probability Measure
Polygon
Tail
Triangle
Valid
Arbitrary

Keywords

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

Cite this

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

In: Probability Theory and Related Fields, Vol. 169, No. 1-2, 01.10.2017, p. 469–521.

Research output: Contribution to journalArticle

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