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July 2015 Branching random tessellations with interaction: A thermodynamic view
Hans-Otto Georgii, Tomasz Schreiber, Christoph Thäle
Ann. Probab. 43(4): 1892-1943 (July 2015). DOI: 10.1214/14-AOP923


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.


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Hans-Otto Georgii. Tomasz Schreiber. Christoph Thäle. "Branching random tessellations with interaction: A thermodynamic view." Ann. Probab. 43 (4) 1892 - 1943, July 2015.


Received: 1 April 2013; Revised: 1 November 2013; Published: July 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1320.60035
MathSciNet: MR3353818
Digital Object Identifier: 10.1214/14-AOP923

Primary: 60D05 , 60K35
Secondary: 28D20 , 60G55 , 82B21

Keywords: Branching tessellation , coloured tessellation , Free energy , Gibbs measure , Relative entropy , STIT tessellation , Stochastic geometry , Variational principle

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 4 • July 2015
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