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.
Highlights
Consider the set of integer points 0 n = {0,. . , n}2 in the n×n square on the plane.A triangulation σ of 0 n is a maximal collection of edges such that each edge has its endpoints in 0 n and, aside from its endpoints, intersects no other edge of σ and no other point ofWe refer to such triangulations as lattice triangulations; see Fig. 1 for an example.Our goal is to study properties of random lattice triangulations
In [5] we introduced a real parameter λ > 0 and studied random lattice triangulations distributed according to the measure λ x∈ n |σx | π(σ ) = Z (λ), σ ∈ n, where |σx | denotes the 1-length of edge σx, and Z (λ) =
Already in the case of one-dimensional, n × 1 lattice triangulations, it is known that properties such as exponential decay of correlations, existence of local limits, tightness of local measures and exponential tails of edge lengths do not hold for any λ ≥ 1
Summary
If we let σ x denote the triangulation obtained from σ by flipping the edge σx , the Markov chain moves from σ to σ x with probability Simulation suggests that this Markov chain has an intriguing behavior, undergoing a phase transition at λ = 1; see Fig. 3. A crucial feature of Theorem 2.3 is that it holds in great generality: in the case of triangulations of general lattice polygons and in the presence of arbitrary constraint edges.2 This generality is key in the application of our main result. The height function is defined in terms of a novel type of geometric crossing, and uses a new partition on the edges of a triangulation in what we call regions of influence In the particular case of triangulations of n ×k rectangles, where k is a fixed integer independent of n, our technique yields stronger results such as the existence of local limits (Theorem 8.4), exponential decay of correlations
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