BRANCHING RANDOM TESSELLATIONS WITH INTERACTION: A THERMODYNAMIC VIEW

• 成果类型: Article
• 署名作者: Georgii, Hans-Otto; Schreiber, Tomasz; Thaele, Christoph
• 署名单位: University of Munich; Nicolaus Copernicus University; Ruhr University Bochum
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP923
• 发表日期: 2015
• 页码: 1892-1943
• 关键词: variational principle; stit tessellations; markov-fields; Iteration; EXISTENCE; geometry

摘要

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