Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

Translation-Invariant p-Adic Gibbs Measures for the Potts Model on the Cayley Tree of Order Four

Authors
Mohammad Azim Mohd Azahari1, *, Mohd Ali Khameini Ahmad1, Nor Muhainiah Mohd Ali1
1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310, Johor Bahru, Johor, Malaysia
*Corresponding author. Email: mohammadazim@graduate.utm.my
Corresponding Author
Mohammad Azim Mohd Azahari
Available Online 12 December 2022.
DOI
10.2991/978-94-6463-014-5_4How to use a DOI?
Keywords
p-adic field; p-adic Gibbs measure; p-adic Potts model; translation-invariant; Cayley trees
Abstract

Different topological structure between real and p-adic fields provides a distinct condition for solution of equations or system of equations. For example, the equation $$x^{2} + 1 = 0$$ does not have solution over real field but it has solution over p-adic field for $$ p \equiv 1\left( {\bmod 4} \right)$$. Meanwhile, the equation $$x^{3} = p$$ has solution in real field but not in p-adic field. It is convenience to investigate the translation-invariant p-adic Gibbs measures of Potts model on Cayley trees in terms of zeros of a certain polynomial. The translation-invariant p-adic Gibbs measures of Potts model on Cayley trees of order two and three was described with respect to some respective conditions on the coefficient of certain quadratic and cubic polynomials. In this paper, the set of p-adic Gibbs measures of p-adic Potts model on the Cayley tree of order four is considered. For this case, it is possible to associate the existence of the translation-invariant p-adic Gibbs measures with zeros of quartic polynomial over p-adic field.

Copyright
© 2023 The Author(s)
Open Access
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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Volume Title
Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)
Series
Advances in Computer Science Research
Publication Date
12 December 2022
ISBN
10.2991/978-94-6463-014-5_4
ISSN
2352-538X
DOI
10.2991/978-94-6463-014-5_4How to use a DOI?
Copyright
© 2023 The Author(s)
Open Access
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

Cite this article

TY  - CONF
AU  - Mohammad Azim Mohd Azahari
AU  - Mohd Ali Khameini Ahmad
AU  - Nor Muhainiah Mohd Ali
PY  - 2022
DA  - 2022/12/12
TI  - Translation-Invariant p-Adic Gibbs Measures for the Potts Model on the Cayley Tree of Order Four
BT  - Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)
PB  - Atlantis Press
SP  - 24
EP  - 33
SN  - 2352-538X
UR  - https://doi.org/10.2991/978-94-6463-014-5_4
DO  - 10.2991/978-94-6463-014-5_4
ID  - Azahari2022
ER  -