Geodesic
Markov processes in the dynamics of primitive triangulations in spaces $R^3$ and $R^4$
Numerical methods and programming, Tome 10 (2009) no. 1, pp. 1-8.

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Lattice models and simplicial complexes continue to play an important role in theoretical physics and gain an increasing interest in connection with the application of dynamic triangulations to the construction of quantum gravity models. With the advent of modern supercomputers, the piecewise-linear complexes and the bistellar transformations become a basis of numerical methods in combinatorial geometry and topology. In this paper, random flips of primitive triangulations in space $R^3$ with vertices from an integer set $Z^3$ are considered as Markov chains and their properties of periodicity, decomposability, and ergodicity are studied. As a result, an asymptotic behavior of the triangulated space as a whole is determined. Similar methods are proposed for primitive triangulations in space $R^4$.
Keywords: primitive triangulation; Diophantine equations; Markov chains; coding of triangulated cubic evolvents; spectrum of vertex polyhedrons; Bose-Einstein statistics. 
@article{VMP_2009_10_1_a0,
     author = {G. G. Ryabov},
     title = {Markov processes in the dynamics of primitive triangulations in spaces $R^3$ and $R^4$},
     journal = {Numerical methods and programming},
     pages = {1--8},
     publisher = {mathdoc},
     volume = {10},
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     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2009_10_1_a0/}
}
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G. G. Ryabov. Markov processes in the dynamics of primitive triangulations in spaces $R^3$ and $R^4$. Numerical methods and programming, Tome 10 (2009) no. 1, pp. 1-8. http://geodesic.mathdoc.fr/item/VMP_2009_10_1_a0/