Existence and simulation of Gibbs-Delaunay-Laguerre tessellations

Daniel Jahn; Filip Seitl

Kybernetika (2020)

  • Volume: 56, Issue: 4, page 617-645
  • ISSN: 0023-5954

Abstract

top
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.

How to cite

top

Jahn, Daniel, and Seitl, Filip. "Existence and simulation of Gibbs-Delaunay-Laguerre tessellations." Kybernetika 56.4 (2020): 617-645. <http://eudml.org/doc/297108>.

@article{Jahn2020,
abstract = {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.},
author = {Jahn, Daniel, Seitl, Filip},
journal = {Kybernetika},
keywords = {Laguerre–Delauay tetrahedrization; stationary Gibbs measure; Gibbs–Laguerre tessellation; MCMC simulation},
language = {eng},
number = {4},
pages = {617-645},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Existence and simulation of Gibbs-Delaunay-Laguerre tessellations},
url = {http://eudml.org/doc/297108},
volume = {56},
year = {2020},
}

TY - JOUR
AU - Jahn, Daniel
AU - Seitl, Filip
TI - Existence and simulation of Gibbs-Delaunay-Laguerre tessellations
JO - Kybernetika
PY - 2020
PB - Institute of Information Theory and Automation AS CR
VL - 56
IS - 4
SP - 617
EP - 645
AB - 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.
LA - eng
KW - Laguerre–Delauay tetrahedrization; stationary Gibbs measure; Gibbs–Laguerre tessellation; MCMC simulation
UR - http://eudml.org/doc/297108
ER -

References

top
  1. Chiu, S. N., Stoyan, D., Kendall, W. S., Mecke, J., 10.1002/9781118658222, J. Willey and Sons, Chichester 2013. MR3236788DOI10.1002/9781118658222
  2. Dereudre, D., 10.1007/978-3-030-13547-8_5, In: Chapter in Stochastic Geometry, pp. 181-229, Springer, Cham 2019. MR3931586DOI10.1007/978-3-030-13547-8_5
  3. Dereudre, D., Drouilhet, R., Georgii, H. O., 10.1007/s00440-011-0356-5, Probab. Theory Rel. 153 (2012), 3, 643-670. MR2948688DOI10.1007/s00440-011-0356-5
  4. Dereudre, D., Lavancier, F., 10.1016/j.csda.2010.05.018, Comput. Stat. Data An. 55 (2011), 1, 498-519. MR2736572DOI10.1016/j.csda.2010.05.018
  5. Fropuff, The vertex configuration of a tetrahedral-octahedral honeycomb. 
  6. Hadamard, P., Résolution d'une question relative aux déterminants., Bull. Sci. Math. 17 (1893), 3, 240-246. 
  7. Lautensack, C., Zuyev, S., 10.1017/s000186780000272x, Adv. Appl. Probab. 40 (2008), 3, 630-650. MR2454026DOI10.1017/s000186780000272x
  8. Møller, J., Waagepetersen, R. P., 10.1201/9780203496930, Chapman and Hall/CRC, Boca Raton 2003. MR2004226DOI10.1201/9780203496930
  9. Okabe, A., Boots, B., Sugihara, K., Chiu, S.N., 10.2307/2687299, J. Willey and Sons, Chichester 2009. MR1770006DOI10.2307/2687299
  10. Preston, C., 10.1007/bfb0080563, Springer, Berlin 1976. MR0448630DOI10.1007/bfb0080563
  11. Quey, R., Renversade, L., 10.1016/j.cma.2017.10.029, Comput. Method Appl. M 330 (2018), 308-333. MR3759098DOI10.1016/j.cma.2017.10.029
  12. Rycroft, C., 10.1063/1.3215722, Chaos 19 (2009), 041111. DOI10.1063/1.3215722
  13. Seitl, F., Petrich, L., Staněk, J., III, C. E. Krill, Schmidt, V., Beneš, V., 10.1007/s11009-019-09757-x, Methodol. Comput. Appl. Probab. (2020). DOI10.1007/s11009-019-09757-x
  14. Sommerville, D. M. Y., An Introduction to the Geometry of N Dimensions., Methuen and Co, London 1929. MR0100239
  15. Stein, P., 10.2307/2315353, Amer. Math. Monthly 73 (1966), 3, 299-301. MR1533698DOI10.2307/2315353
  16. Zessin, H., 10.3103/s11957-008-1005-x, J. Contemp. Math. Anal. 43 (2008), 1, 59-65. MR2465001DOI10.3103/s11957-008-1005-x

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.