Rectangular Gilbert Tessellation

TL;DR

This paper introduces a rectangular Gilbert tessellation model, analyzes the distribution of segment lengths, and proves exponential decay of correlations, providing bounds and insights into the structure of the tessellation.

Contribution

It extends Gilbert tessellation to a rectangular case, deriving exponential bounds for segment length distribution and correlation decay, with new results on boundary behavior.

Findings

Exponential bounds for the tail of segment length distribution.

Exponential decay of correlations between segments.

Linear growth of rays reaching the boundary in a confined box.

Abstract

A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of rays from a given set of points. This model turns out to be a particular (rectangular) case of the well-known Gilbert tessellation, which originally models the growth of needle-shaped crystals from the initial random points with a Poisson distribution in a plane. From each point the lines grow on both sides of vertical and horizontal directions until they meet another line. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segments are proved to decay exponentially with the distance between their initial points. Furthermore, the sign of the correlation is investigated for some instructive examples. In the case when the initial set of points is confined in a box $[0,N]^2$, it is proved that the average number of rays reaching the border of the box has a linear order in $N$.