Packing Density of Combinatorial Settlement Planning Models

AbstractWe consider a combinatorial settlement model on a rectangular grid where each house must be exposed to sunlight from east, south, or west. We are interested in maximal configurations, where no additional houses can be added. Once the settlement is completely built, it seems natural to consider the building density of the obtained maximal configuration. In this article we consider two different random models which produce maximal configurations and, using simulations, we plot an estimate of the distribution of the building density (actually, the occupancy—the total number of houses built) and we conjecture that the means of these distributions converge to a certain limit as the grid dimensions grow to infinity.MSC: 60C0590C27 AcknowledgmentsWe thank the anonymous referees for helpful comments that have led to improvements of the presentation of the article. We also wish to thank Professors Tomislav Došlić and Pavel Krapivsky for fruitful and stimulating discussions.Notes1 Our Southern Hemisphere friends are welcome to turn the page upside down when inspecting the figures in our paper.2 We write X=(d)Y if two random variables X and Y are equal in distribution. Since we are dealing with discrete random variables, this is the same as requiring P(X=z)=P(Y=z) for all z∈R.Additional informationNotes on contributorsMate PuljizMATE PULJIZ was born in Croatia in 1988. He received his master’s degree from the University of Zagreb, Croatia, in 2012 and his Ph.D. in Pure Mathematics from the University of Birmingham, United Kingdom, in 2017. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include abstract dynamical systems, particularly the topology of their orbits, particle models, and fractional calculus. mate.puljiz@fer.hrStjepan ŠebekSTJEPAN ŠEBEK was born in Croatia in 1990. He received his master’s degree in 2014 and his Ph.D. in Mathematics in 2019, both from the University of Zagreb, Croatia. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include geometry and potential theory of random walks.Josip ŽubrinićJOSIP ŽUBRINIĆ was born in Croatia in 1993. He received his master’s degree in 2016 and his Ph.D. in Mathematics in 2022, both from the University of Zagreb, Croatia. He is currently a Postdoc with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include homogenization and dimension reduction in the theory of elasticity. josip.zubrinic@fer.hr