Rectangular Gilbert Tessellation

Abstract: A random planar quadrangulation process is introduced as an approximation for certain cellular automata describing neuronal dynamics. This model turns out to be a particular (rectangular) case of a well-known Gilbert tessellation. The central and still open question is the distribution of the line segments. We derive exponential bounds for the tail of this distribution. The correlations in the model are also investigated for some instructive examples. In particular, the exponential decay of correlations is established. If the initial configuration of points is confined in a box $[0,N]<sup>2$ it is proved that the average number of rays escaping the box has a linear order in $N$.