Topological properties of a random partition of the plane

My Random partitions of the plane ...... paper describes the partitions (within a certain model) as measurable sets, but their topological properties are unclear. In particular, do the boundaries have fractal dimension greater than 1?

Update (June 2018). John Preater kindly informed me of his 2009 paper A species of voter model driven by immigration which treats a very similar model and proves the analog of my Conjecture 3, that the topological boundary of each region has Lebesgue measure zero.

Update (July 2019). Simulations by Marc Barthelemy estimate the fractal dimension of the boundaries at about 1.26. Also he has simulations of the distribution of areas of countries in the self-similar limit (with average area equals one). One's first guess might be a log-Normal distribution, but this graphic shows that is a poor fit.

Other aspects of the self-similar limit partition involve the network whose vertices are the capital cities (a rate-1 Poisson point process) and whose edges indicate that the countries are neighbors (share a border). This is in principle not planar but simulations and intuition suggest crossing edges are very rare. See this simulation with 100 countries, and compare with e.g. the Gabriel network. Our network has a more heterogeneous look because of greater variability of areas. The physics literature discusses several empirically observed "laws" for random tessellations (Aboav-Weaire's and Lewis' laws) involving areas and degrees (number of neighbors). It is not clear whether one should expect these to hold in our fractal setting. Here is some data.

The first two are roughly consistent with the Lewis law but the third is somewhat different (concave rather than convex) from the Aboav law.

Back to Open Problem list