Geodesic
Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 1, pp. 156-176 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order $k\geq 2$. For the Potts model in a translation-invariant external field for $k\geq 2$, the nonuniqueness of the translation-invariant and periodic Gibbs measure is shown. It is proved that for the Potts model in an external field that is not translation-invariant, translation-invariant Gibbs measures do not exist on the Cayley tree of order $k\geq 2$. Periodic Gibbs measures are also studied for the Potts model in a periodic external field. We prove that under certain conditions, the number of such measures can be at least three.
Keywords: Cayley tree, Gibbs measure, Potts model, periodic external field, translation-invariant external field. 
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U. A. Rozikov; M. M. Rahmatullaev; R. M. Khakimov. Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 1, pp. 156-176. http://geodesic.mathdoc.fr/item/TMF_2022_210_1_a9/

[1] Kh.-O. Georgi, Gibbsovskie mery i fazovye perekhody, Mir, M., 1992 | MR | Zbl

[2] K. Preston, Gibbsovskie sostoyaniya na schetnykh mnozhestvakh, Mir, M., 1977 | MR

[3] Ya. G. Sinai, Teoriya fazovykh perekhodov. Strogie rezultaty, Nauka, M., 1980 | MR | MR | Zbl

[4] N. N. Ganikhodzhaev, “Gruppovoe predstavlenie i avtomorfizmy dereva Keli”, Dokl. AN RUz, 1994, no. 4, 3–5

[5] N. N. Ganikhodzhaev, U. A. Rozikov, “Opisanie periodicheskikh krainikh gibbsovskikh mer nekotorykh reshetochnykh modelei na dereve Keli”, TMF, 111:1 (1997), 109–117 | DOI | DOI | MR | Zbl

[6] N. N. Ganikhodzhaev, U. A. Rozikov, “Gruppovoe predstavlenie lesa Keli i ego nekotorye primeneniya”, Izv. RAN. Ser. matem., 67:1 (2003), 21–32 | DOI | DOI | MR | Zbl

[7] N. N. Ganikhodjaev, F. M. Mukhamedov, J. F. F. Mendes, “On the three state Potts model with competing interactions on the Bethe lattice”, J. Stat. Mech., 2006:8 (2006), P08012, 29 pp. | DOI

[8] L. V. Bogachev, U. A. Rozikov, “On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field”, J. Stat. Mech., 2019:7 (2019), 073205, 76 pp. | DOI

[9] U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore, 2013 | DOI | MR

[10] P. M. Blekher, N. N. Ganikhodzhaev, “O chistykh fazakh modeli Izinga na reshetkakh Bete”, Teoriya veroyatn. i ee primen., 35:2 (1990), 220–230 | DOI | MR | Zbl

[11] P. M. Bleher, J. Ruiz, V. A. Zagrebnov, “On the purity of the limiting Gibbs states for the Ising model on the Bethe lattice”, J. Stat. Phys., 79:1–2 (1995), 473–482 | DOI

[12] P. M. Bleher, J. Ruiz, V. A. Zagrebnov, “On the phase diagram of the random field Ising model on the Bethe lattice”, J. Stat. Phys., 93:1–2 (1998), 33–78 | DOI

[13] N. N. Ganikhodzhaev, “O chistykh fazakh ferromagnitnoi modeli Pottsa s tremya sostoyaniyami na reshetke Bete vtorogo poryadka”, TMF, 85:2 (1990), 163–175 | DOI | MR

[14] N. N. Ganikhodzhaev, “O chistykh fazakh ferromagnitnoi modeli Pottsa na reshetke Bete”, Dokl. AN RUz, 1992, no. 6–7, 4–7

[15] N. N. Ganikhodjaev, U. A. Rozikov, “The Potts model with countable set of spin values on a Cayley tree”, Lett. Math. Phys., 75:2 (2006), 99–109 | DOI

[16] C. Külske, U. A. Rozikov, R. M. Khakimov, “Description of translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree”, J. Stat. Phys., 156:1 (2014), 189–200, arXiv: 1310.622 | DOI

[17] U. A. Rozikov, R. M. Khakimov, F. Kh. Khaidarov, “Krainost translyatsionno-invariantnykh mer Gibbsa dlya modeli Pottsa na dereve Keli”, TMF, 196:1 (2018), 117–134 | DOI | DOI | MR

[18] C. Külske, U. A. Rozikov, “Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree”, Random Struct. Algor., 50:4 (2017), 636–678 | DOI

[19] M. M. Rakhmatullaev, “Clabo periodicheskie mery Gibbsa i osnovnye sostoyaniya dlya modeli Pottsa s konkuriruyuschimi vzaimodeistviyami na dereve Keli”, TMF, 176:3 (2013), 477–493 | DOI | DOI | MR | Zbl

[20] M. M. Rakhmatullaev, “Suschestvovanie slabo periodicheskikh mer Gibbsa dlya modeli Pottsa na dereve Keli”, TMF, 180:3 (2014), 307–317 | DOI | DOI | MR

[21] M. M. Rakhmatullaev, “O slabo periodicheskikh merakh Gibbsa dlya modeli Pottsa s vneshnim polem na dereve Keli”, Ukr. matem. zhurn., 68:4 (2016), 529–541

[22] M. M. Rahmatullaev, “On weakly periodic Gibbs measures of the Potts model with a special external field on a Cayley tree”, Zhurn. matem. fiz., anal., geom., 12:4 (2016), 302–314 | DOI | MR

[23] U. A. Rozikov, “Gibbs measures of Potts model on Cayley trees: A survey and applications”, Rev. Math. Phys., 33 (2021), 2130007, 58 pp. | DOI

[24] U. A. Rozikov, R. M. Khakimov, “Periodicheskie mery Gibbsa dlya modeli Pottsa na dereve Keli”, TMF, 175:2 (2013), 300–312 | DOI | DOI | MR | Zbl

[25] R. M. Khakimov, F. Kh. Khaidarov, “Translyatsionno-invariantnye i periodicheskie mery Gibbsa dlya modeli Pottsa na dereve Keli”, TMF, 189:2 (2016), 286–295 | DOI | DOI | MR