Branching random tessellations with interaction: A thermodynamic view

TL;DR

This paper introduces a new class of branching random tessellations with interaction, extending STIT tessellations by allowing cell interactions and colours, and develops a thermodynamic framework including entropy, energy, and variational principles.

Contribution

It generalizes STIT tessellations to include interactions and colours, providing a Gibbsian framework and existence results for these complex tessellations.

Findings

Defined a division kernel characterizing cell interactions.

Established a variational principle for translation invariant BRTs.

Proved existence of BRTs with prescribed division kernels.

Abstract

A branching random tessellation (BRT) is a stochastic process that transforms a coarse initial tessellation of $\mathbb{R}^d$ into a finer tessellation by means of random cell divisions in continuous time. This concept generalises the so-called STIT tessellations, for which all cells split up independently of each other. Here, we allow the cells to interact, in that the division rule for each cell may depend on the structure of the surrounding tessellation. Moreover, we consider coloured tessellations, for which each cell is marked with an internal property, called its colour. Under a suitable condition, the cell interaction of a BRT can be specified by a measure kernel, the so-called division kernel, that determines the division rules of all cells and gives rise to a Gibbsian characterisation of BRTs. For translation invariant BRTs, we introduce an "inner" entropy density relative to a STIT tessellation. Together with an inner energy density for a given "moderate" division kernel, this leads to a variational principle for BRTs with this prescribed kernel, and further to an existence result for such BRTs.