Paper Info
Title
A Lyapunov function for Glauber dynamics on lattice triangulations.
Abstract
We study random triangulations of the integer points \([0,n]^2 \cap {\mathbb {Z}}^2\), where each triangulation \(\sigma \) has probability measure \(\lambda ^{|\sigma |}\) with \(\lambda >0\) being a real parameter and \(|\sigma |\) denoting the sum of the length of the edges in \(\sigma \). Such triangulations are called lattice triangulations. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime \(\lambda <1\), 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 \(\lambda <1\). 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 \(\lambda <1\). 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.
Year
DOI
Venue
2015
10.1007/s00440-016-0735-z
Probability Theory and Related Fields
Keywords
Field
DocType
Lattice triangulations, Glauber dynamics, Lyapunov function, Primary 60J10, Secondary 60K35, 52C20, 05C10
Integer,Discrete mathematics,Lyapunov function,Glauber,Combinatorics,Exponential function,Lattice (order),Probability measure,Triangulation (social science),Mathematics,Lambda
Journal
Volume
Issue
ISSN
abs/1504.07980
1-2
1432-2064
Citations 
PageRank 
References 
0
0.34
1
Authors
1
Name
Order
Citations
PageRank
Alexandre O. Stauffer113011.34