A Lyapunov function for Glauber dynamics on lattice triangulations

Abstract

We study random triangulations of the integer points [0, n] 2 ∩ Z 2 , where each triangulation σ has probability measure λ |σ| with |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.1 The set of vertices Λ 0 n does not need to be the n × n square, but can be the set of integer points inside any, even non-convex, lattice polygon (a polygon whose vertices are points of Z 2 ).2 Triangulations where some given set of edges are forced to be present, see Section 2 for precise definitions.

Spectral Properties of Unimodular Lattice Triangulations

“…There are exact enumeration results and analytical bounds [43] as well as numerical approximations of this number [45] using Monte-Carlo simulations, both show that there are exponentially many triangulations in terms of the system size M × N. So lattice triangulations are extensive and can be treated as a well-defined statistical system. Additionally, the convergence of Glauber dynamics on lattice triangulations was examined in [15,68] for different parameter sets.…”

“…A similar problem was considered analytically for lattice triangulations in Refs. [15,68], where Glauber dynamics was used, which is basically the Metropolis algorithm with a slightly different choice of the acceptance probabilities. Therein, the authors used an energy function that measures the sum of the edge lengths that qualitatively agrees with the energy function (2).…”

Dynamics of lattice triangulations on thin rectangles

“…The proof of Theorem 1.1 will rely crucially on some recent developments by one of us [13] based on a Lyapunov function approach to the sub-critical regime λ ∈ (0, 1). As detailed in subsequent sections, the main results of [13] will be used first to show that after T = O(n 2 ) steps of the chain we can reduce the problem to a restricted chain on a "good" set of triangulations, each edge of which never exceeds logarithmic length, and then to show that distant regions in our thin rectangles can be decoupled with an exponentially small error. This will enable us to set up a recursive scheme for functional inequalities related to mixing time such as the logarithmic Sobolev inequality.…”

“…This allows one to obtain an upper bound on the relaxation time of a Markov chain in terms of the congestion ratio restricted to a subspace Ω and the time the chain needs to visit Ω with large probability. Here we use a further crucial input from [13] permitting us to identify a "canonical" subset of triangulations Ω such that after T = O(n 2 ) the chain enters Ω with large probability and such that the chain restricted to Ω has small congestion ratio. A detailed high-level overview of the proof will be given in Section 4.1.…”

“…The rest of the paper is organized as follows. In Section 2, we first recall some important tools from [3] and then formulate the main ingredients we need from [13]. Then, in Section 3 we develop the applications of improved canonical path techniques to our setting.…”