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AN IMPROVED LOWER BOUND FOR THE CRITICAL PARAMETER OF STAVSKAYA’S PROCESS

Part of: Special processes

Published online by Cambridge University Press:  13 May 2020

ALEX D. RAMOS Open the ORCID record for ALEX D. RAMOS [Opens in a new window] ,
CALITÉIA S. SOUSA Open the ORCID record for CALITÉIA S. SOUSA [Opens in a new window] ,
PABLO M. RODRIGUEZ Open the ORCID record for PABLO M. RODRIGUEZ [Opens in a new window]  and
PAULA CADAVID Open the ORCID record for PAULA CADAVID [Opens in a new window]
ALEX D. RAMOS
Affiliation:
Department of Statistics, Universidade Federal de Pernambuco, Recife, PE, 50740-540, Brazil email [email protected]
CALITÉIA S. SOUSA
Affiliation:
Department of Statistics, Universidade Federal de Pernambuco, Recife, PE, 50740-540, Brazil email [email protected]
PABLO M. RODRIGUEZ*
Affiliation:
Department of Statistics, Universidade Federal de Pernambuco, Recife, PE, 50740-540, Brazil email [email protected]
PAULA CADAVID
Affiliation:
Universidade Federal do ABC, Santo André, SP, 09210-580, Brazil email [email protected]
*
✉[email protected]
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Abstract

We consider Stavskaya’s process, which is a two-state probabilistic cellular automaton defined on a one-dimensional lattice. The state of any vertex depends only on itself and on the state of its right-adjacent neighbour. This process was one of the first multicomponent systems with local interaction for which the existence of a kind of phase transition has been rigorously proved. However, the exact localisation of its critical value remains as an open problem. We provide a new lower bound for the critical value.

Keywords

particle random processone-dimensional local interactionphase transitionStavskaya’s processprobabilistic cellular automata

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 517 - 524
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This work has been partially supported by FAPESP (2017/10555-0), CNPq (Grant 304676/2016-0) and CAPES (under the Program MATH-AMSUD/CAPES 88881.197412/2018-01).

References

Depoorter, J. and Maes, C., ‘Stavskaya’s measure is weakly Gibbsian’, Markov Process. Related Fields 12 (2006), 791–804.Google Scholar
Gantmacher, F. R., Applications of the Theory of Matrices (Interscience Publishers, New York, 1959).Google Scholar
Harris, T. E., ‘Contact interactions on a lattice’, Ann. Probab. 2(6) (1974), 969–988.CrossRefGoogle Scholar
Liggett, T. M., Interacting Particle Systems (Springer, Berlin, 1985).CrossRefGoogle Scholar
Mendonça, J., ‘Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton’, Phys. Rev. E 83(1) (2011), 012102.Google ScholarPubMed
Ponselet, L., Phase Transitions in Probabilistic Cellular Automata, PhD Thesis, Université catholique de Louvain, 2013.Google Scholar
Taggi, L., ‘Critical probabilities and convergence time of percolation probabilistic cellular automata’, J. Stat. Phys. 159(4) (2015), 853–892.CrossRefGoogle Scholar
Toom, A. L., ‘A family of uniform nets of formal neurons’, Sov. Math. Dokl. 9(6) (1968).Google Scholar
Toom, A., Contornos, Conjuntos Convexos e Autômatos Celulares (in Portuguese), 23o Colóquio Brasileiro de Matemática (IMPA, Rio de Janeiro, 2001).Google Scholar
Toom, A., Ergodicity of Cellular Automata, Tartu University, Estonia, 2013. Available online at http://math.ut.ee/emsdk/intensiivkursused/TOOM-TARTU-3.pdf.Google Scholar
Toom, A., Vasilyev, N., Stavskaya, O., Mityushin, L., Kurdyumov, G. and Pirogov, S., ‘Discrete local Markov systems’, in: Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, Nonlinear Science: Theory and Applications (eds. Dobrushin, R., Kryukov, V. and Toom, A.) (Manchester University Press, Manchester, 1990).Google Scholar