Dynamics of Lattice Triangulations on Thin Rectangles

TL;DR

This paper investigates the mixing times of lattice triangulations on thin rectangles, revealing a phase transition at a critical parameter value, with polynomial bounds for certain regimes and exponential lower bounds for others.

Contribution

It establishes a dynamical phase transition in the mixing times of lattice triangulations on thin rectangles at the critical point , providing tight bounds for <<1.

Findings

Polynomial mixing time bounds for <<1

Exponential lower bounds for >1

Existence of a phase transition at =1

Abstract

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $\lambda^{|\sigma|}$ where $\lambda>0$ is a parameter and $|\sigma|$ denotes the total edge length of the triangulation. When $\lambda\in(0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp(\Omega(n^2))$ for $\lambda>1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $\lambda=1$.