Abstract
We study random triangulations of the integer points [0,n]2∩Z2 , where each triangulation σ has probability measure λσ 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 language  English 

Pages (fromto)  469–521 
Number of pages  53 
Journal  Probability Theory and Related Fields 
Volume  169 
Issue number  12 
Early online date  10 Aug 2016 
DOIs  
Publication status  Published  31 Oct 2017 
Keywords
 math.PR
 cs.DM
 mathph
 math.CO
 math.MP
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Alexandre Stauffer
Person: Research & Teaching