Branching random tessellations with interaction: A thermodynamic view

• Georgii, Hans-Otto ^[2] ; Schreiber, Tomasz ^[3] ; Thäle, Christoph ^[1]
  1. [1] Ruhr University Bochum

     Ruhr University Bochum

     Kreisfreie Stadt Bochum, Alemania

     
  2. [2] Ludwig Maximilian University of Munich

     Ludwig Maximilian University of Munich

     Kreisfreie Stadt München, Alemania

     
  3. [3] Nicolaus Copernicus University

     Nicolaus Copernicus University

     Toruń, Polonia

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 4, 2015, págs. 1892-1943
• Idioma: inglés
• DOI: 10.1214/14-AOP923
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• Resumen

  • A branching random tessellation (BRT) is a stochastic process that transforms a coarse initial tessellation of Rd 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.