×

Periodic \(p\)-adic Gibbs measures of \(q\)-state Potts model on Cayley trees. I: The chaos implies the vastness of the set of \(p\)-adic Gibbs measures. (English) Zbl 1397.82008

The authors aim to contribute to the \(p\)-adic counterpart of the theory of Gibbs measures on Cayley trees. They generalize the results obtained by U. A. Rozikov and O. N. Khakimov [Markov Process. Relat. Fields 21, No. 1, 177–204 (2015; Zbl 1332.82018)] and M. Saburov and M. A. Khameini Ahmad [Math. Phys. Anal. Geom. 18, No. 1, Article ID 26, 33 p. (2015; Zbl 1456.82181)]. The authors investigate the set of \(p\)-adic Gibbs measures of the \(q\)-state Potts model on the Cayley tree of order three. They obtain new results under the conditions \(|\theta-1|_p <1\) and \(0<|q|_p <1\), where \(\theta = \exp_p(J )\) and \(J\) is a coupling constant. They prove the vastness of the set of (periodic) \(p\)-adic Gibbs measures by showing the chaotic behavior of the corresponding Potts-Bethe mapping over the field \(\mathbb{Q}_p\) for primes \(p \equiv 1\;(\text{mod}\;3)\) with the condition \(0<|\theta-1|_p <|q|_p<1\). The authors obtain a new family of the periodic \(p\)-adic Gibbs measure of the \(q\)-state Potts model by means of the \(H_m\)-periodic function. They prove that, for a semi-infinite Cayley tree, there exist many periodic \(p\)-adic Gibbs measures of periods other than 1 and 2. The authors then point out a subsystem that is isometrically conjugate to the shift dynamics. The dynamics of the Potts-Bethe mapping over the fields \(\mathbb{Q}_2\) and \(\mathbb{Q}_3\) are discussed. Lastly, the authors show that, except for a fixed point and the inverse images of the singular point, all points converge to an attracting fixed point.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C05 Trees
81Q65 Alternative quantum mechanics (including hidden variables, etc.)
82B26 Phase transitions (general) in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albeverio, S., Cianci, R., Khrennikov, A.Yu.: \(p\)-adic valued quantization. p-Adic Num. Ultra. Anal. Appl. 1(2), 91-104 (2009) · Zbl 1187.81137
[2] Albeverio, S., Cianci, R., Khrennikov, A.Yu.: On the Fourier transform and the spectral properties of the \(p\)-adic momentum and Schrodinger operators. J. Phys. A Math. Gen. 30, 5767-5784 (1997) · Zbl 0927.46060
[3] Albeverio, S., Cianci, R., Khrennikov, A.Yu.: A representation of quantum field Hamiltonian in a \(p\)-adic Hilbert space. Theor. Math. Phys. 112(3), 355-374 (1997) · Zbl 0968.46519
[4] Albeverio, S., Cianci, R., Khrennikov, A.Yu.: On the spectrum of the \(p\)-adic position operator. J. Phys. A Math. Gen. 30, 881-889 (1997) · Zbl 0992.81022
[5] Albeverio, S., Khrennikov, A.Yu., Shelkovich, V.M.: Theory of p-Adic Distributions: Linear and Nonlinear Models. Cambridge University Press, Cambridge (2010) · Zbl 1198.46001
[6] Beltrametti, E; Cassinelli, G, Quantum mechanics and \(p\)-adic numbers, Found. Phys., 2, 1-7, (1972) · doi:10.1007/BF00708614
[7] Borevich, Z.I., Shafarevich, I.R.: Number Theory. Acad Press, New York (1966) · Zbl 0145.04902
[8] Dragovich, B., Khrennikov, A.Yu., Kozyrev, S.V., Volovich, I.V.: On \(p\)-adic mathematical physics. p-Adic Num. Ultra. Anal. Appl. 1(1), 1-17 (2009) · Zbl 1187.81004
[9] Dragovich, B., Khrennikov, A.Yu., Kozyrev, S.V., Volovich, I.V., Zelenov, E.I.: \(p\)-Adic mathematical physics: the first 30 years. p-Adic Num. Ultra. Anal. Appl. 9(2), 87-121 (2017) · Zbl 1394.81009
[10] Gandolfo, D., Maes, C., Ruiz, J., Shlosman, S.: Glassy states: the free Ising model on a tree. arXiv:1709.00543 (2017)
[11] Gandolfo, D; Rahmatullaev, MM; Rozikov, UA, Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley tree, J. Stat. Phys., 167, 1164-1179, (2017) · Zbl 1376.82027 · doi:10.1007/s10955-017-1771-5
[12] Gandolfo, D; Ruiz, J; Shlosman, S, A manifold of pure Gibbs states of the Ising model on a Cayley tree, J. Stat. Phys., 148, 999-1005, (2012) · Zbl 1254.82008 · doi:10.1007/s10955-012-0574-y
[13] Ganikhodjaev, N; Mukhamedov, F; Rozikov, U, Existence of a phase transition for the Potts \(p\)-adic model on the set \(\mathbb{Z}\), Theor. Math. Phys., 130, 425-431, (2002) · Zbl 1031.82013 · doi:10.1023/A:1014723108030
[14] Georgii, H.O.: Gibbs Measures and Phase Transitions. W. de Gruyter, Berlin (2011) · Zbl 1225.60001 · doi:10.1515/9783110250329
[15] Ilic-Stepic, A; Ognjanovic, Z; Ikodinovic, N; Perovic, A, A \(p\)-adic probability logic, Math. Log. Q., 58, 263-280, (2012) · Zbl 1251.03027 · doi:10.1002/malq.201110006
[16] Ilic-Stepic, A; Ognjanovic, Z; Ikodinovic, N, Conditional \(p\)-adic probability logic, Int. J. Approx. Reas., 55, 1843-1865, (2014) · Zbl 1433.03062 · doi:10.1016/j.ijar.2014.02.001
[17] Ilic-Stepic, A; Ognjanovic, Z, Logics for reasoning about processes of thinking with information coded by \(p\)-adic numbers, Stud. Log., 103, 145-174, (2015) · Zbl 1382.03045 · doi:10.1007/s11225-014-9552-5
[18] Ilic-Stepic, A; Ognjanovic, Z; Ikodinovic, N; Perovic, A, \(p\)-adic probability logics, p-Adic Num. Ultra. Anal. Appl., 8, 177-203, (2016) · Zbl 1353.03013 · doi:10.1134/S2070046616030018
[19] Koblitz, N.: p-Adic Numbers, p-Adic Analysis, and Zeta Functions. Springer, New York (1984) · Zbl 0364.12015 · doi:10.1007/978-1-4612-1112-9
[20] Khrennikov, A.Yu.: Non-Archimedean white noise. In: Proc. Int. Conf. on Gaussian Random Fields, Nagoya, 127 (1990) · Zbl 1286.82006
[21] Khrennikov, A.Yu.: Mathematical methods of non-Archimedean physics. Rus. Math. Surv. 45, 87-125 (1990) · Zbl 0722.46040
[22] Khrennikov, A.Yu.: \(p\)-adic quantum mechanics with \(p\)-adic valued wave functions. J. Math. Phys. 32, 932-937 (1991) · Zbl 0746.46067
[23] Khrennikov, A.Yu.: \(p\)-adic statistic and probability. Dokl. Acad. Nauk. SSSR 322(6), 1075-1079 (1992) · Zbl 1096.82007
[24] Khrennikov, A.Yu.: Axiomatics of the \(p\)-adic theory of probabilities. Dokl. Acad. Nauk. SSSR 326(5), 1075-1079 (1992)
[25] Khrennikov, A.Yu.: \(p\)-adic probability theory and its applications. A principle of the statistical stabilization of frequencies. Theor. Math. Phys. 97(3), 348-363 (1993) · Zbl 0839.60005
[26] Khrennikov, A.Yu.: p-Adic Valued Distributions in Mathematical Physics. Kluwer, Dordrecht (1994) · Zbl 0833.46061
[27] Khrennikov, A.Yu.: Non-Archimedean theory of probability: frequency and axiomatic theories. Acta Math. Appl. Sin. 12(1), 78-92 (1996) · Zbl 0863.60006
[28] Khrennikov, A.Yu.: \(p\)-adic valued probability measures. Indag. Math. N. S. 7(3), 311-330 (1996) · Zbl 0872.60002
[29] Khrennikov, A.Yu.: Interpretations of Probability. Walter de Gruyter, Berlin (2009) · Zbl 1369.81014
[30] Khrennikov, A.Yu., Ludkovsky, S.: On infinite products of non-Archimedean measure spaces. Indag. Math. N. S. 13(2), 177-183 (2002) · Zbl 1017.60015
[31] Khrennikov, A.Yu., Yamada, Sh., van Rooij, A.: The measure-theoretical approach to \(p\)-adic probability theory. Ann. Math. Blaise Pascal 6(1), 21-32 (1999) · Zbl 0941.60010
[32] Kingsbery, J; Levin, A; Preygel, A; Silva, CE, On measure-preserving \(\cal{C}^{1}\) transformations of compact-open subsets of non-Archimedean local fields, Trans. Am. Math. Soc., 361, 61-85, (2009) · Zbl 1155.37003 · doi:10.1090/S0002-9947-08-04686-2
[33] Kulske, C; Rozikov, UA; Khakimov, RM, Description of all translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree, J. Stat. Phys., 156, 189-200, (2013) · Zbl 1298.82022 · doi:10.1007/s10955-014-0986-y
[34] Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995) · Zbl 1106.37301 · doi:10.1017/CBO9780511626302
[35] Ludkovsky, S., Khrennikov, A.Yu.: Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields. Markov Process. Relat. Fields 9, 131-162 (2003) · Zbl 1017.60045
[36] Fan, AH; Liao, LM; Wang, YF; Zhou, D, \(p\)-adic repeller in \(\mathbb{Q}_p\) are subshifts of finite type, C. R. Math. Acad. Sci. Paris, 344, 219-224, (2007) · Zbl 1108.37016 · doi:10.1016/j.crma.2006.12.007
[37] Fan, AH; Fan, SL; Liao, LM; Wang, YF, On minimal decomposition of \(p\)-adic homographic dynamical systems, Adv. Math., 257, 92-135, (2014) · Zbl 1392.37102 · doi:10.1016/j.aim.2014.02.007
[38] Mezard, M., Parisi, G., Virasoro, M.: Spin-Glass: Theory and Beyond. World Scientific, Singapore (1987) · Zbl 0992.82500
[39] Mukhamedov, F, On dynamical systems and phase transitions for \((q+1)\)-state \(p\)-adic Potts model on the Cayley tree, Math. Phys. Anal. Geom., 16, 49-87, (2013) · Zbl 1280.46047 · doi:10.1007/s11040-012-9120-z
[40] Mukhamedov, F., Akin, H.: Phase transitions for \(p\)-adic Potts model on the Cayley tree of order three. J. Stat. Mech. P07014 (2013) · Zbl 1286.82004
[41] Mukhamedov, F; Khakimov, O, Phase transition and chaos: \(p\)-adic Potts model on a Cayley tree, Chaos Solit. Fract., 87, 190-196, (2016) · Zbl 1355.37106 · doi:10.1016/j.chaos.2016.04.003
[42] Mukhamedov, F; Khakimov, O, On periodic Gibbs measures of \(p\)-adic Potts model on a Cayley tree, p-Adic Num. Ultra. Anal. Appl., 8, 225-235, (2016) · Zbl 1356.82022 · doi:10.1134/S2070046616030043
[43] Mukhamedov, F; Khakimov, O, On Julia set and chaos in \(p\)-adic Ising model on the Cayley tree, Math. Phys. Anal. Geom., 20, 23, (2017) · Zbl 1413.46067 · doi:10.1007/s11040-017-9254-0
[44] Mukhamedov, F; Khakimov, O, Chaotic behavior of the \(p\)-adic Potts-Bethe mapping, Discret. Contin. Dyn. Syst., 38, 231-245, (2018) · Zbl 1372.37022 · doi:10.3934/dcds.2018011
[45] Mukhamedov, F., Khakimov, O.: Chaotic behavior of the \(p\)-adic Potts-Bethe mapping II. (preprint) · Zbl 1372.37022
[46] Mukhamedov, F; Omirov, B; Saburov, M, On cubic equations over \(p\)-adic field, Int. J. Number Theory, 10, 1171-1190, (2014) · Zbl 1307.11124 · doi:10.1142/S1793042114500201
[47] Mukhamedov, F; Omirov, B; Saburov, M; Masutova, K, Solvability of cubic equations in \(p\)-adic integers, \(p>3\), Sib. Math. J., 54, 501-516, (2013) · Zbl 1294.11205 · doi:10.1134/S0037446613030154
[48] Mukhamedov, F; Saburov, M, On equation \(x^q=a\) over \(\mathbb{Q}_p\), J. Number Theory, 133, 55-58, (2013) · Zbl 1293.11062 · doi:10.1016/j.jnt.2012.07.006
[49] Mukhamedov, F., Saburov, M., Khakimov, O.: On \(p\)-adic Ising-Vannimenus model on an arbitrary order Cayley tree, J. Stat. Mech. P05032 (2015) · Zbl 1342.82041
[50] Mukhamedov, F; Rozikov, U, On Gibbs measures of \(p\)-adic Potts model on Cayley tree, Indag. Math. N. S., 15, 85-100, (2004) · Zbl 1161.82311 · doi:10.1016/S0019-3577(04)90007-9
[51] Mukhamedov, F; Rozikov, U, On inhomogeneous \(p\)-adic Potts model on a Cayley tree, Infin. Dimen. Anal. Quantum. Probab. Relat. Top., 8, 277-290, (2005) · Zbl 1096.82007 · doi:10.1142/S0219025705001974
[52] Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974) · Zbl 0297.60054 · doi:10.1017/CBO9780511897122
[53] Rozikov, U; Khakimov, O, Description of all translation-invariant \(p\)-adic Gibbs measures for the Potts model on a Cayley tree, Markov Process. Relat. Fields, 21, 177-204, (2015) · Zbl 1332.82018
[54] Rozikov, U; Khakimov, O, \(p\)-adic Gibbs measures and Markov random fields on countable graphs, Theor. Math. Phys., 175, 518-525, (2013) · Zbl 1286.82013 · doi:10.1007/s11232-013-0042-0
[55] Rozikov, UA; Khakimov, RM, Periodic Gibbs measures for the Potts model on the Cayley tree, Theor. Math. Phys., 175, 699-709, (2013) · Zbl 1286.82006 · doi:10.1007/s11232-013-0055-8
[56] Rozikov, U, Representability of trees and some of their applications, Math. Notes, 72, 479-488, (2002) · Zbl 1028.82008 · doi:10.1023/A:1020580227830
[57] Rozikov, U.: Gibbs Measures on Cayley Trees. World Sci. Pub, Singapore (2013) · Zbl 1278.82002 · doi:10.1142/8841
[58] Rozikov, U, Gibbs measures on Cayley trees: results and open problems, Rev. Math. Phys., 25, 1330001, (2013) · Zbl 1268.60122 · doi:10.1142/S0129055X1330001X
[59] Saburov, M., Ahmad, M.A.Kh.: Solvability criteria for cubic equations over \({\mathbb{Z}}_2^{*}\). AIP Conf. Proc. 1602, 792-797 (2014) · Zbl 07753048
[60] Saburov, M; Ahmad, MAKh, Solvability of cubic equations over \({\mathbb{Q}}_3\), Sains Malays., 44, 635-641, (2015) · Zbl 1332.11100 · doi:10.17576/jsm-2015-4404-20
[61] Saburov, M., Ahmad, M.A.Kh.: The number of solutions of cubic equations over \({\mathbb{Q}}_3\). Sains Malays. 44(5), 765-769 (2015) · Zbl 1332.11101
[62] Saburov, M., Ahmad, M.A.Kh.: Quadratic equations over \(p\)-adic fields and their application in statistical mechanics. Sci. Asia 41(3), 209-215 (2015)
[63] Saburov, M., Ahmad, M.A.Kh.: On descriptions of all translation invariant \(p\)-adic Gibbs measures for the Potts model on the Cayley tree of order three. Math. Phys. Anal. Geom. 18, 26 (2015) · Zbl 1456.82181
[64] Saburov, M., Ahmad, M.A.Kh.: Solvability and number of roots of bi-quadratic equations over \(p\)-adic fields. Malays. J. Math. Sci. 10, 15-35 (2016)
[65] Saburov, M., Ahmad, M.A.Kh.: Local descriptions of roots of cubic equations over \(p\)-adic fields. Bull. Malays. Math. Sci. Soc. 41, 965-984 (2018) · Zbl 1427.11128
[66] Saburov, M., Ahmad, M.A.Kh.: The dynamics of the Potts-Bethe mapping over \({\mathbb{Q}}_p\): the case \(p≡ \) 2 (mod 3). J. Phys. 819(1) (2017)
[67] Silverman, J.: The Arithmetic of Dynamical Systems. Springer, New York (2007) · Zbl 1130.37001 · doi:10.1007/978-0-387-69904-2
[68] Spitzer, F, Markov random field on infinite tree, Ann. Prob., 3, 387-398, (1975) · Zbl 0313.60072 · doi:10.1214/aop/1176996347
[69] Vladimirov, V.S., Volovich, I.V., Zelenov, E.V.: p-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994) · Zbl 0812.46076 · doi:10.1142/1581
[70] Volovich, IV, \(p\)-adic strings, Class. Quantum Grav., 4, 83-87, (1987) · doi:10.1088/0264-9381/4/4/003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.