Geodesic
Translation-invariant $p$-adic quasi Gibbs measures for the Potts model with an external field on the Cayley tree
Nanosistemy: fizika, himiâ, matematika, Tome 16 (2025) no. 2, pp. 164-175 Cet article a éte moissonné depuis la source Math-Net.Ru

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The study is focused on investigation of $p$-adic Gibbs measures for the $q$-state Potts model with an external field and determination of the conditions for the existence of a phase transition. In this work, we derive a functional equation that satisfies the compatibility condition for $p$-adic quasi-Gibbs measures on a Cayley tree of order $k\ge$ 2. Furthermore, we prove that if $|q|_p$ = 1 there exists a unique $p$-adic Gibbs measure for this model. Additionally, for the Potts model on a binary tree, we identify three $p$-adic quasi-Gibbs measures under specific circumstances: one bounded and two unbounded, which implies a phase transition.
Keywords: $p$-adic numbers, the Potts model with external field, $p$-adic quasi Gibbs measure, translationinvariant, Cayley tree. 
@article{NANO_2025_16_2_a4,
     author = {Muzaffar M. Rahmatullaev and Nurkhon D. Samijonova},
     title = {Translation-invariant $p$-adic quasi {Gibbs} measures for the {Potts} model with an external field on the {Cayley} tree},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
     pages = {164--175},
     year = {2025},
     volume = {16},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/NANO_2025_16_2_a4/}
}
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Muzaffar M. Rahmatullaev; Nurkhon D. Samijonova. Translation-invariant $p$-adic quasi Gibbs measures for the Potts model with an external field on the Cayley tree. Nanosistemy: fizika, himiâ, matematika, Tome 16 (2025) no. 2, pp. 164-175. http://geodesic.mathdoc.fr/item/NANO_2025_16_2_a4/