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Laguerre–Delauay tetrahedrization; stationary Gibbs measure; Gibbs–Laguerre tessellation; MCMC simulation
Three-dimensional Laguerre tessellation models became quite popular in many areas of physics and biology. They are generated by locally finite configurations of marked points. Randomness is included by assuming that the set of generators is formed by a marked point process. The present paper focuses on 3D marked Gibbs point processes of generators which enable us to specify the desired geometry of the Laguerre tessellation. In order to prove the existence of a stationary Gibbs measure using a general approach of Dereudre, Drouilhet and Georgii [3], the geometry of Laguerre tessellations and their duals Laguerre Delaunay tetrahedrizations is examined in detail. Since it is difficult to treat the models analytically, their simulations are carried out by Markov chain Monte Carlo techniques.
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