Review Paper – Review Papers and Research Papers containing a Review sectionNo Access

Gibbs measures of Potts model on Cayley trees: A survey and applications

U. A. Rozikov

U. A. Rozikov

V. I. Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences, Uzbekistan

AKFA University, 1st Deadlock 10, Kukcha Darvoza, Tashkent 100095, Uzbekistan

Faculty of Mathematics, National University of Uzbekistan, Uzbekistan

E-mail Address: rozikovu@yandex.ru

Search for more papers by this author

https://doi.org/10.1142/S0129055X21300077Cited by:7 (Source: Crossref)

Abstract

In this paper, we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc.

Keywords:

AMSC: 82B20, 82B26

References

1. K. Adam, D. Zöllner and D. P. Field, 3D microstructural evolution of primary recrystallization and grain growth in cold rolled single-phase aluminum alloys, Model. Simul. Mater. Sci. Eng. 26 (2018) 035011, 16 pp. Crossref, Web of Science, Google Scholar
2. R. Allena, M. Scianna and L. Preziosi, A cellular Potts model of single cell migration in presence of durotaxis, Math. Biosci. 275 (2016) 57–70. Crossref, Web of Science, Google Scholar
3. É. O. Aquino, F. W. Lima, A. D. Araújo and F. R. N. Costa, Potts model in one-dimension on directed small-world networks, J. Stat. Phys. 171(6) (2018) 1112–1121. Crossref, Web of Science, ADS, Google Scholar
4. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989), Reprint of the 1982 original. Google Scholar
5. L. Beaudin, A review of the Potts model, Rose-Hulman Undergraduate Math. J. 8(1) (2007) Article 13, 25 pp. Google Scholar
6. L. Beaudin, J. Ellis-Monaghan, G. Pangborn and R. Shrock, A little statistical mechanics for the graph theorist, Discrete Math. 310(13–14) (2010) 2037–2053. Crossref, Web of Science, Google Scholar
7. P. M. Bleher, J. Ruiz and V. A. Zagrebnov, On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice, J. Stat. Phys. 79 (1995) 473–482. Crossref, Web of Science, ADS, Google Scholar
8. P. Bleher, M. Lyubich and R. Roeder, Lee–Yang–Fisher zeros for the DHL and 2D rational dynamics, II. Global pluripotential interpretation, J. Geom. Anal. 30(1) (2020) 777–833. Crossref, Web of Science, Google Scholar
9. S. E. M. Boas, Y. Jiang, R. M. H. Merks, S. A. Prokopiou and E. G. Rens, Cellular Potts model: Applications to vasculogenesis and angiogenesis, in Probabilistic Cellular Automata, Emergence, Complexity and Computation, Vol. 27 (Springer, Cham, 2018), pp. 279–310. Crossref, Google Scholar
10. L. V. Bogachev and U. A. Rozikov, On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field, J. Stat. Mech. Theory Exp. 7 (2019) 073205, 76 pp. Google Scholar
11. G. I. Botirov and U. A. Rozikov, The Potts model with competing interactions on the Cayley tree: The contour method, Theor. Math. Phys. 153(1) (2007) 1423–1433. Crossref, Web of Science, Google Scholar
12. T. Brookings, J. M. Carlson and J. Doyle, Three mechanisms for power laws on the Cayley tree, Phys. Rev. E 72 (2005) 056120, 18 pp. Crossref, Web of Science, Google Scholar
13. F. Caltagirone, G. Parisi and T. Rizzo, Dynamical critical exponents for the mean-field Potts glass, Phys. Rev. E 85 (2012) 051504. Crossref, Web of Science, ADS, Google Scholar
14. J. Cavender, Taxonomy with confidence, Math. BioSci. 40 (1978) 271–280. Crossref, Web of Science, Google Scholar
15. S. Chatterjee, M. Mangeat, R. Paul and H. Rieger, Flocking and reorientation transition in the 4-state active Potts model, Europhys. Lett. 130(6) (2020) 66001, 8 pp. Crossref, Google Scholar
16. G. Celeux, F. Forbes and N. Peyrard, EM-based image segmentation using Potts models with external field, [Research Report] RR-4456, INRIA (2002), inria-00072132. Google Scholar
17. L. Cioletti and R. Vila, Graphical representations for Ising and Potts models in general external fields, J. Stat. Phys. 162(1) (2016) 81–122. Crossref, Web of Science, ADS, Google Scholar
18. S. Cohen-Addad, R. Höhler and O. Pitois, Flow in foams and flowing foams, Annu. Rev. Fluid Mech. 45 (2013) 241–267. Crossref, Web of Science, ADS, Google Scholar
19. L. Coquille, H. Duminil-Copin, D. Ioffe and Y. Velenik, On the Gibbs states of the noncritical Potts model on $ℤ^{2}$ , Probab. Theory Relat. Fields 158(1–2) (2014) 477–512. Crossref, Web of Science, Google Scholar
20. L. Coquille, Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions, J. Stat. Phys. 159(4) (2015) 958–971. Crossref, Web of Science, ADS, Google Scholar
21. C. Daskalakis, E. Mossel and S. Roch, Evolutionary trees and the Ising model on the Bethe lattice: A proof of Steel’s conjecture, Probab. Theory Relat. Fields 149(1–2) (2011) 149–189. Crossref, Web of Science, Google Scholar
22. A. Dembo, A. Montanari, A. Sly and N. Sun, The replica symmetric solution for Potts models on $d$ -regular graphs, Comm. Math. Phys. 327 (2014) 551–575. Crossref, Web of Science, ADS, Google Scholar
23. H. Duminil-Copin, V. Sidoravicius and V. Tassion, Continuity of the phase transition for planar random-cluster and Potts models with $1 \leq q \leq 4$ , Comm. Math. Phys. 349(1) (2017) 47–107. Crossref, Web of Science, ADS, Google Scholar
24. O. N. Feldheim and Y. Spinka, Long-range order in the 3-state antiferromagnetic Potts model in high dimensions, J. Eur. Math. Soc. 21(5) (2019) 1509–1570. Crossref, Web of Science, Google Scholar
25. M. Formentin and C. Külske, A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton–Watson trees, Electron. Commun. Probab. 14 (2009) 587–596. Web of Science, Google Scholar
26. G. Franzese and A. Coniglio, Phase transitions in the Potts spin-glass model, Phys. Rev. E 58(3) (1998) 2753–2759. Crossref, Web of Science, ADS, Google Scholar
27. S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction (Cambridge University Press, Cambridge, 2018). Google Scholar
28. D. Gandolfo, M. M. Rakhmatullaev, U. A. Rozikov and J. Ruiz, On free energies of the Ising model on the Cayley tree, J. Stat. Phys. 150(6) (2013) 1201–1217. Crossref, Web of Science, ADS, Google Scholar
29. D. Gandolfo, M. M. Rahmatullaev and U. A. Rozikov, Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley trees, J. Stat. Phys. 167(5) (2017) 1164–1179. Crossref, Web of Science, ADS, Google Scholar
30. N. N. Ganikhodjaev, The Potts model on $ℤ^{d}$ with countable set of spin values, J. Math. Phys. 45(3) (2004) 1121–1127. Crossref, Web of Science, ADS, Google Scholar
31. N. N. Ganikhodjaev, On pure phases of the three-state ferromagnetic Potts model on the second-order Bethe lattice, Theor. Math. Phys. 85(2) (1990) 1125–1134. Crossref, Web of Science, Google Scholar
32. N. N. Ganikhodzhaev, On pure phases of the ferromagnetic Potts model Bethe lattices, Dokl. AN Uzbekistan 6-7 (1992) 4–7. Google Scholar
33. N. N. Ganikhodjaev and U. A. Rozikov, The Potts model with countable set of spin values on a Cayley tree, Lett. Math. Phys. 75(2) (2006) 99–109. Crossref, Web of Science, ADS, Google Scholar
34. N. N. Ganikhodzhaev and U. A. Rozikov, Description of periodic extreme Gibbs measures of some lattice models on a Cayley tree, Theor. Math. Phys. 111(1) (1997) 480–486. Crossref, Web of Science, Google Scholar
35. A. Galanis, D. Štefankovič, E. Vigoda and L. Yang, Ferromagnetic Potts model: Refined #BIS-hardness and related results, SIAM J. Comput. 45(6) (2016) 2004–2065. Crossref, Web of Science, Google Scholar
36. H. O. Georgii, Gibbs Measures and Phase Transitions, 2nd edn., de Gruyter Studies in Mathematics, Vol. 9 (Walter de Gruyter, Berlin, 2011). Crossref, Google Scholar
37. F. Graner and J. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett. 69(13) (1992) 2013–2017. Crossref, Web of Science, ADS, Google Scholar
38. W. M. Haddad, A Dynamical Systems Theory of Thermodynamics, Princeton Series in Applied Mathematics (Princeton University Press, Princeton, NJ, 2019). Google Scholar
39. O. Häggström and C. Külske, Gibbs properties of the fuzzy Potts model on trees and in mean field, Markov Proc. Rel. Fields 10(3) (2004) 477–506. Google Scholar
40. Y. Higuchi, Remarks on the limiting Gibbs states on a $(d + 1)$ -tree, Publ. RIMS, Kyoto Univ. 3 (1977) 335–348. Crossref, Google Scholar
41. M. A. Idiart and Y. Levin, Rupture of a liposomal vesicle, Phys. Rev. E 69(6) (2004) 061922. Crossref, Web of Science, ADS, Google Scholar
42. I. Kanter, Potts-glass models of neural networks, Phys. Rev. A (3) 37(7) (1988) 2739–2742. Crossref, Web of Science, ADS, Google Scholar
43. A. Karolak, D. A. Markov, L. J. McCawley and K. A. Rejniak, Towards personalized computational oncology: From spatial models of tumour spheroids, to organoids, to tissues, J. R. Soc. Interface 15 (2018) 20170703, 16 pp. Crossref, Web of Science, Google Scholar
44. H. Kesten and B. P. Stigum, Additional limit theorem for indecomposable multi-dimensional Galton-Watson processes, Ann. Math. Statist. 37 (1966) 1463–481. Crossref, Google Scholar
45. R. M. Khakimov, On the existence of periodic Gibbs measures for the Potts model on a Cayley tree, Uzbek. Mat. Zh. 3 (2014) 134–142 (in Russian). Google Scholar
46. R. M. Khakimov, New periodic Gibbs measures for q-state Potts model on a Cayley tree, J. Siberian Federal Univ. Math. Phys. 7(3) (2014) 297–304. Google Scholar
47. R. M. Khakimov and M. T. Makhammadaliev, Translation invariance of periodic Gibbs measures for the Potts model on the Cayley tree, Theor. Math. Phys. 199(2) (2019) 726–735. Crossref, Web of Science, Google Scholar
48. R. M. Khakimov and F. Kh. Khaidarov, Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree, Theor. Math. Phys. 189(2) (2016) 1651–1659. Crossref, Web of Science, Google Scholar
49. E. Kropff and A. Treves, The storage capacity of Potts models for semantic memory retrieval, J. Stat. Mech. Theory Exp. 8 (2005) P08010, 19 pp. Google Scholar
50. C. Külske, U. A. Rozikov and R. M. Khakimov, Description of the translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree, J. Stat. Phys. 156(1) (2014) 189–200. Crossref, Web of Science, ADS, Google Scholar
51. C. Külske and 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. Crossref, Web of Science, Google Scholar
52. C. Külske, D. Meiβner, Stable and metastable phases for the Curie–Weiss–Potts model in vector-valued fields via singularity theory, J. Stat. Phys. 181(3) (2020) 968–989. Crossref, Web of Science, ADS, Google Scholar
53. R. M. Levy, A. Haldane and W. F. Flynn, Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary fitness, Curr. Opin. Struct. Biol. 43 (2017) 55–62. Crossref, Web of Science, Google Scholar
54. Q. Li, X. Wang, F. Liang, F. Yi, Y. Xie, A. Gazdar and G. Xiao, A Bayesian hidden Potts mixture model for analyzing lung cancer pathology images, Biostatistics 20(4) (2019) 565–581. Crossref, Web of Science, Google Scholar
55. D. M. Lopes and J. C. M. Mombach, Two-dimensional wetting transition modeling with the Potts model, Braz. J. Phys. 47 (2017) 672–677. Crossref, Web of Science, ADS, Google Scholar
56. M. Löwe and F. Vermet, The capacity of $q$ -state Potts neural networks with parallel retrieval dynamics, Statist. Probab. Lett. 77(14) (2007) 1505–1514. Crossref, Web of Science, Google Scholar
57. M. Naim, V. Boboeva, C. J. Kang and A. Treves, Reducing a cortical network to a Potts model yields storage capacity estimates, J. Stat. Mech. Theory Exp. 4 (2018) 043304, 35 pp. Google Scholar
58. L. Nicolao and M. Ostilli, Critical states in political trends. How much reliable is a poll on Twitter?, Phys. A 533 (2019) 121920, 18 pp. Crossref, Web of Science, Google Scholar
59. E. P. Normatov and U. A. Rozikov, A description of harmonic functions using the properties of the group representation of the Cayley tree, Math. Notes 79(3–4) (2006) 399–407. Crossref, Web of Science, Google Scholar
60. M. Ostilli and F. Mukhamedov, 1D three-state mean-field Potts model with first- and second-order phase transitions, Phys. A 555 (2020) 124415, 10 pp. Crossref, Web of Science, Google Scholar
61. M. Mak, F. Spill, K. Roger and M. Zaman, Single-cell migration in complex microenvironments: Mechanics and signaling dynamics, J. Biomech. Eng. 138(2) (2016) 021004, 8 pp. Crossref, Web of Science, Google Scholar
62. E. Marinari, S. Mossa and G. Parisi, Glassy Potts model: A disordered Potts model without a ferromagnetic phase, Phys. Rev. B 59 (1999) 8401. Crossref, Web of Science, ADS, Google Scholar
63. J. B. Martin, Reconstruction thresholds on regular trees, in Discrete Random Walks (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc., Vol. AC (Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003), pp. 191–204. Crossref, Google Scholar
64. P. P. Martin and S. F. Zakaria, Zeros of the 3-state Potts model partition function for the square lattice revisited, J. Stat. Mech. Theory Exp. 8 (2019) 084003, 25 pp. Google Scholar
65. F. Martinelli, A. Sinclair and D. Weitz, Fast mixing for independent sets, coloring and other models on trees, Random Struct. Algor. 31 (2007) 134–172. Crossref, Web of Science, Google Scholar
66. H. Meyer-Ortmanns, Immigration, integration and ghetto formation, Int. J. Mod. Phys. C 14(3) (2003) 311–320. Link, Web of Science, ADS, Google Scholar
67. E. Mossel, Reconstruction on trees: Beating the second eigenvalue, Ann. Appl. Probab. 11(1) (2001) 285–300. Crossref, Web of Science, Google Scholar
68. E. Mossel and Y. Peres, Information flow on trees, Ann. Appl. Probab. 13(3) (2003) 817–844. Crossref, Web of Science, Google Scholar
69. E. Mossel, Survey: Information flow on trees, in Graphs, Morphisms and Statistical Physics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 63 (American Mathematical Society, Providence, RI, 2004), pp. 155–170. Crossref, Google Scholar
70. E. Mossel, On the impossibility of reconstructing ancestral data and phylogenies, J. Comput. Biol. 10(5) (2003) 669–678. Crossref, Web of Science, Google Scholar
71. E. Mossel and M. Steel, A phase transition for a random cluster model on phylogenetic trees, Math. Biosci. 187(2) (2004) 189–203. Crossref, Web of Science, Google Scholar
72. F. M. Mukhamedov, U. A. Rozikov and J. F. Mendes, On contour arguments for the three state Potts model with competing interactions on a semi-infinite Cayley tree, J. Math. Phys. 48(1) (2007) 013301, 14 pp. Crossref, Web of Science, Google Scholar
73. F. M. Mukhamedov and U. A. Rozikov, On Gibbs measures of $p$ -adic Potts model on the Cayley tree, Indag. Math., New Ser. 15 (2004) 85–100. Crossref, Web of Science, Google Scholar
74. F. M. Mukhamedov and U. A. Rozikov, On inhomogeneous $p$ -adic Potts model on a Cayley tree, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005) 277–290. Link, Web of Science, Google Scholar
75. F. M. Mukhamedov, U. A. Rozikov and J. F. F. Mendes, On phase transitions for $p$ -adic Potts model with competing interactions on a Cayley tree, in p-Adic Mathematical Physics: 2nd International Conference, AIP Conference Proceedings, Vol. 826 (Amer. Inst. Phys., Melville, NY, 2006), pp. 140–150. Crossref, Google Scholar
76. F. M. Mukhamedov, O. N. Khakimov, On periodic Gibbs measures of $p$ -adic Potts model on a Cayley tree, p-Adic Numbers Ultrametric Anal. Appl. 8(3) (2016) 225–235. Crossref, Web of Science, Google Scholar
77. F. M. Mukhamedov, O. N. Khakimov, Phase transition and chaos: $P$ -adic Potts model on a Cayley tree, Chaos Solitons Fractals 87 (2016) 190–196. Crossref, Web of Science, ADS, Google Scholar
78. E. Müller-Hartmann, Theory of the Ising model on a Cayley tree, Z. Phys. B 27 (1977) 161–168. Crossref, ADS, Google Scholar
79. D. Panchenko, Free energy in the Potts spin glass, Ann. Probab. 46(2) (2018) 829–864. Crossref, Web of Science, Google Scholar
80. U. Paun, G method in action: Normalization constant, important probabilities, and fast exact sampling for Potts model on trees, Rev. Roumaine Math. Pures Appl. 65(2) (2020) 101–128. Web of Science, Google Scholar
81. F. Peruggi, F. di Liberto, G. Monroy, The Potts model on Bethe lattices. I. General results, J. Phys. A 16(4) (1983) 811–827. Crossref, ADS, Google Scholar
82. F. Peruggi, F. di Liberto, G. Monroy, Phase diagrams of the $q$ -state Potts model on Bethe lattices, Phys. A 141(1) (1987) 151–186. Crossref, Web of Science, Google Scholar
83. I. J. Polmear, Light Alloys from Traditional Alloys to Nanocrystals, 4th edn. (Elsevier Butterworth-Heinemann, Burlington, MA, 2006). Google Scholar
84. R. B. Potts, Some generalized order-disorder transformations, Proc. Cambridge Philos. Soc. 48 (1952) 106–109. Crossref, Web of Science, ADS, Google Scholar
85. C. Preston, Gibbs States on Countable Sets (Cambridge University Press, London, 1974). Crossref, Google Scholar
86. B. Radhakrishnan, G. B. Sarma, T. Zacharia, Modeling the kinetics and microstructural evolution during static recrystallization-Monte Carlo simulation of recrystallization, Acta Mater. 46 (1998) 4415–4433. Crossref, Web of Science, ADS, Google Scholar
87. M. M. Rahmatullaev, The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree, Theor. Math. Phys. 180(3) (2014) 1019–1029. Crossref, Web of Science, Google Scholar
88. M. M. Rahmatullaev, A weakly periodic Gibbs measure for the ferromagnetic Potts model on a Cayley tree, Sib. Math. J. 56(5) (2015) 929–935. Crossref, Web of Science, Google Scholar
89. M. M. Rahmatullaev, On weakly periodic Gibbs measures of the Potts model with a special external field on a Cayley tree, Zh. Mat. Fiz. Anal. Geom. 12(4) (2016) 302–314. Crossref, Google Scholar
90. M. M. Rahmatullaev, On weakly periodic Gibbs measures for the Potts model with an external field on a Cayley tree, Ukrainian Math. J. 68(4) (2016) 598–611. Crossref, Web of Science, Google Scholar
91. M. A. Rasulova, On periodic Gibbs measures for the Potts-SOS model on a Cayley tree, Theor. Math. Phys. 199(1) (2019) 586–592. Crossref, Web of Science, Google Scholar
92. J. Reichardt and S. Bornholdt, Detecting fuzzy community structures in complex networks with a Potts model, Phys. Rev. Lett. 93 (2004) Article ID: 218701, 4 pp. Crossref, Web of Science, Google Scholar
93. N. F. Robertson, M. Pawelkiewicz, J. L. Jacobsen and H. Saleur, Integrable boundary conditions in the antiferromagnetic Potts model, J. High Energy Phys. 5 (2020) 144, 34 pp. Google Scholar
94. H. E. Rodriguez, P. Barrat-Charlaix and M. Weigt, Toward inferring Potts models for phylogenetically correlated sequence data, Entropy 21(11) (2019) Paper No. 1090, 20 pp. Web of Science, Google Scholar
95. B. Rosental, Z. Kozhekbaeva, N. Fernhoff, J. M. Tsai and N. Traylor-Knowles, Coral cell separation and isolation by fluorescence-activated cell sorting, BMC Cell Biol. 18 (2017) 30. Crossref, Google Scholar
96. U. A. Rozikov, On pure phase of the anti-ferromagnetic Potts model on the Cayley tree, Uzbek Math. J. 1 (1999) 73–77 (in Russian). Google Scholar
97. U. A. Rozikov, Description uncountable number of Gibbs measures for inhomogeneous Ising model, Theor. Math. Phys. 188(1) (1999) 77–84. Crossref, Web of Science, Google Scholar
98. U. A. Rozikov, On q-component models on Cayley tree: Contour method, Lett. Math. Phys. 71(1) (2005) 27–38. Crossref, Web of Science, ADS, Google Scholar
99. U. A. Rozikov and Y. M. Suhov, Gibbs measures for SOS models on a Cayley tree, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 471–488. Link, Web of Science, Google Scholar
100. U. A. Rozikov, A constructive description of ground states and Gibbs measures for Ising model with two-step interactions on Cayley tree, J. Stat. Phys. 122(2) (2006) 217–235. Crossref, Web of Science, ADS, Google Scholar
101. U. A. Rozikov, A contour method on Cayley trees, J. Stat. Phys. 130(4) (2008) 801–813. Crossref, Web of Science, ADS, Google Scholar
102. U. A. Rozikov and F. T. Ishankulov, Description of periodic $p$ -harmonic functions on Cayley trees, Nonlinear Diff. Equations Appl. 17(2) (2010) 153–160. Crossref, Web of Science, Google Scholar
103. U. A. Rozikov, Gibbs Measures on Cayley Trees (World Scientific Publisher, Singapore, 2013). Link, Google Scholar
104. U. A. Rozikov, R. M. Khakimov, Periodic Gibbs measures for the Potts model on the Cayley tree, Theor. Math. Phys. 175(2) (2013) 699–709. Crossref, Web of Science, Google Scholar
105. U. A. Rozikov, Gibbs measures on Cayley trees: Results and open problems, Rev. Math. Phys. 25(1) (2013) Article ID: 1330001, 112 pp. Link, Web of Science, Google Scholar
106. U. A. Rozikov, Tree-hierarchy of DNA and distribution of Holliday junctions, J. Math. Biol. 75(6–7) (2017) 1715–1733. Crossref, Web of Science, Google Scholar
107. U. A. Rozikov and M. M. Rahmatullaev, On free energies of the Potts model on the Cayley tree, Theor. Math. Phys. 190(1) (2017) 98–108. Crossref, Web of Science, Google Scholar
108. U. A. Rozikov and G. I. Botirov, Non-translation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree, Rep. Math. Phys. 81(1) (2018) 105–115. Crossref, Web of Science, Google Scholar
109. U. A. Rozikov, Holliday junctions for the Potts model of DNA. in Algebra, Complex Analysis, and Pluripotential Theory, eds. Z. Ibragimov et al., Springer Proceedings in Mathematics and Statistics, Vol. 264 (Springer, Cham, 2018), pp. 151–165. Crossref, Google Scholar
110. U. A. Rozikov, R. M. Khakimov and F. Kh. Khaidarov, Extremality of translation invariant Gibbs measures for Potts model on Cayley tree, Theor. Math. Phys. 196(1) (2018) 1043–1058. Crossref, Web of Science, Google Scholar
111. U. A. Rozikov, Thermodynamics of interacting system of DNAs, Theor. Math. Phys. 206(2) (2021) 174–183. Crossref, Web of Science, Google Scholar
112. U. A. Rozikov, Thermodynamics of DNA-RNA renaturation, Int. J. Geom. Methods Mod. Phys. 18(6) (2021) Article ID: 2150096, 14 pp. Link, Web of Science, Google Scholar
113. S. Sanyal and J. A. Glazier, Viscous instabilities in flowing foams: A Cellular Potts Model approach, J. Stat. Mech. Theory Exp. 10 (2006) P10008, 9 pp. Google Scholar
114. J. Salas and A. D. Sokal, Absence of phase transitions for the antiferromagnetic Potts model via the Dobrushin uniqueness theorem, J. Stat. Phys. 86 (1997) 551–579. Crossref, Web of Science, ADS, Google Scholar
115. T. C. Schelling, Dynamic models of segregation, J. Math. Sociol. 1 (1971) 143–186. Crossref, Web of Science, Google Scholar
116. C. Schulze, Potts-like model for ghetto formation in multi-cultural societies, Int. J. Mod. Phys. C 16(3) (2005) 351–355. Link, Web of Science, ADS, Google Scholar
117. A. Shahraz, A. Borhan and K. A. Fichthorn, Wetting on physically patterned solid surfaces: The relevance of molecular dynamics simulations to macroscopic systems, Langmuir 29(37) (2013) 11632–11639. Crossref, Web of Science, Google Scholar
118. A. Shahraz, A. Borhan and K. A. Fichthorn, Kinetics of droplet wetting mode transitions on grooved surfaces: Forward flux sampling, Langmuir 30(51) (2014) 15442–15450. Crossref, Web of Science, Google Scholar
119. A. N. Shiryaev, Probability, 2nd edn., Graduate Texts in Mathematics, Vol. 95 (Springer, New York, 1996). Crossref, Google Scholar
120. Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results (Pergamon, Oxford 1982). Google Scholar
121. T. N. de Silva and V. Sivised, A statistical mechanics perspective for protein folding from $q$ -state Potts model, preprint (2017), arXiv:1709.04813. Google Scholar
122. A. Sly, Reconstruction for the Potts model, Ann. Probab. 39(4) (2011) 1365–1406. Crossref, Web of Science, Google Scholar
123. J. Srolvitz, M. P. Anderson, G. S. Grest and P. S. Sahni, Grain growth in two-dimensions, Scr. Metall. 17 (1983) 241–246. Crossref, Web of Science, Google Scholar
124. M. Steel and M. Charleston, Five surprising properties of parsimoniously colored trees, Bull. Math. Biol. 57 (1995) 367–375. Crossref, Web of Science, Google Scholar
125. L. Sun, Y. F. Chang and X. Cai, A discrete simulation of tumor growth concerning nutrient concentration, Int. J. Mod. Phys. B 18 (2004) 2651–2657. Link, Web of Science, ADS, Google Scholar
126. A. Szabó and R. M. H. Merks, Cellular Potts modeling of tumor growth, tumor invasion, and tumor evolution, Front. Oncol. 3 (2013) Article ID: 87, 12 pp. Crossref, Google Scholar
127. T. Takaishi, Simulations of financial markets in a Potts-like model, Int. J. Mod. Phys. C 16(08) (2005) 1311–1317. Link, Web of Science, ADS, Google Scholar
128. D. R. Tan, C. D. Li, W. P. Zhu and F. J. Jiang, A comprehensive neural networks study of the phase transitions of Potts model, New J. Phys. 22 (2020) 063016, 18 pp. Crossref, Web of Science, Google Scholar
129. V. Tikare and J. D. Cawley, Application of the Potts model to simulation of Ostwald ripening, J. Amer. Ceramic Soc. 81(3) (1998) 485–491. Crossref, Web of Science, Google Scholar
130. P. N. Timonin, Statistics of geometric clusters in Potts model: Statistical mechanics approach, Proc. A 476(2240) (2020) 215–228. Google Scholar
131. A. Voss-Böhme, Cellular Potts models for interacting cell populations: Mathematical foundation, challenges, and future prospects, in Probabilistic Cellular Automata, Emergence, Complexity and Computation, Vol. 27 (Springer, Cham, 2018), pp. 311–325. Crossref, Google Scholar
132. M. J. de Vries, Technological processes, in Teaching About Technology, Contemporary Issues in Technology Education (Springer, 2016). Crossref, Google Scholar
133. J. Wang, J. Wang and H. E. Stanley, Multiscale multifractal DCCA and complexity behaviors of return intervals for Potts price model, Phys. A 492 (2018) 889–902. Crossref, Web of Science, Google Scholar
134. D. Weaire, F. Bolton, P. Molho and J. A. Glaziers, Investigation of an elementary model for magnetic froth, J. Phys.: Condens. Matter 3(13) (1991) 2101–2115. Crossref, Web of Science, ADS, Google Scholar
135. F. Y. Wu, The Potts model, Rev. Mod. Phys. 54 (1982) 235–268. Crossref, Web of Science, ADS, Google Scholar
136. D. Xiong and H. Zhao, $q$ -state Potts-glass neural network based on pseudoinverse rule, J. Phys. A 43(44) (2010) 445001, 9 pp. Crossref, Google Scholar
137. K. Xu and J. Wang, Weighted fractional permutation entropy and fractional sample entropy for nonlinear Potts financial dynamics, Phys. Lett. A 381(8) (2017) 767–779. Crossref, Web of Science, ADS, Google Scholar