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<p>A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of lines, called 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 rays grow on both sides of vertical and horizontal directions until they meet another ray. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of the line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segment lengths 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\([0,N]^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, it is proved that the average number of rays reaching the border of the box has a linear order in <i>N</i>.</p>

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Rectangular Gilbert Tessellation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月13日

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  • Emily Ewers,
  • Tatyana S. Turova

摘要

A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of lines, called 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 rays grow on both sides of vertical and horizontal directions until they meet another ray. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of the line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segment lengths 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 \([0,N]^2\) [ 0 , N ] 2 , it is proved that the average number of rays reaching the border of the box has a linear order in N.