Periodic Gibbs measures for the soft-core Widom–Rowlinson model and their extremality
DOI:
https://doi.org/10.56143/ujmcs.v2i3s.10Keywords:
Cayley tree, Widom–Rowlinson model, Gibbs-measure, translation-invariant Gibbs measure, periodic Gibbs measure, extremality, nonextremality, phase transition.Abstract
We consider both hard-core and soft-core Widom–Rowlinson models with spin values \((-1, 0, 1)\) on a Cayley tree of order \(k \geq 2\), focusing on the Gibbs measures of these models. The models depend on three parameters: the order \(k\) of the tree, the interaction strength \(\theta\) (describing \(J < 0\) and \(J > 0\) cases), and the particle intensity \(\lambda\). The hard-core Widom–Rowlinson model corresponds to the case \(\theta = 0\). For \(k = 2\) and \(J > 0\), we prove the existence of two and four non-translation-invariant (non-TI) periodic Gibbs measures within the corresponding regions of the parameters \(\theta\) and \(\lambda\). Since the explicit forms of the four solutions to the system of functional equations are obtained, the conditions for the extremality and non-extremality of these periodic Gibbs measures are established. In the case \(k = 3\) and \(J > 0\), under the condition \(\theta > 2\), explicit expressions for two critical values \(\lambda_{c,t,i}(\theta)\) (\(i = 1, 2\)) are derived. Using these critical values, we show that there exist exactly two or exactly four non-translation-invariant periodic Gibbs measures in the corresponding regions of the parameter space.
References
[1] Bleher, P. M., Ganikhodjaev, N. N.: On pure phases of the Ising model on the Bethe lattice. Theor. Probab. Appl. 35, 216–227 (1990).
[2] Bleher, P. M., Ruiz, J., Zagrebnov, V. A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79, 473–482 (1995).
[3] Higuchi, Y.: Remarks on the limiting Gibbs states on a (d + 1)-tree. Publ. Res. Inst. Math. Sci. 13, 335–348 (1977).
[4] Chayes, J. T., Chayes, L., Kotecký, R.: The analysis of the Widom-Rowlinson model by stochastic geometric methods. Comm. Math. Phys. 172, 551–569 (1995).
[5] Formentin, M., Külske, C.: Symmetric Entropy Bound on the Non-Reconstruction Regime of Markov Chains on Galton-Watson Trees. Electron. Comm. Probab. 14, 587–596 (2009).
[6] Gallavotti, G., Lebowitz, J.: Phase Transitions in Binary Lattice Gases. J. Math. Phys. 12, 1129–1133 (1971).
[7] Georgii, H. O.: Gibbs Measures and Phase Transitions. Second edition, de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin (2011).
[8] Higuchi, Y., Takei, M.: Some results on the phase structure of the two-dimensional Widom-Rowlinson model. Osaka J. Math. 41, 237–255 (2004).
[9] Jahnel, B., Külske, C.: The Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27, 3845–3892 (2017).
[10] Ganikhodjaev, N. N., Rozikov, U. A.: Description of periodic extreme Gibbs measures of some lattice models on a Cayley tree. Theor. Math. Phys. 111 (1), 480–486 (1997).
[11] Külske, C., Rozikov, U. A.: Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree. Random Struct. Alg. 50 (4), 636–678 (2017).
[12] Martin, J. B., Rozikov, U. A., Suhov, Y. M.: A three state hard-core model on a Cayley tree. Jour. Nonlinear Math. Phys. 12, 432-448 (2005). [13] Mazel, A., Suhov, Y., Stuhl, I., Zohren, S.: Dominance of most tolerant species in multi-type lattice Widom-Rowlinson models. J. Stat. Mech. 2014, (2014).
[14] Mazel, A., Stuhl, I., Suhov, Y.: Hard-core configurations on a triangular lattice and Eisenstein primes. Preprint arXiv:1803.04041.
[15] Rozikov, U. A., Suhov, Y. M.: Gibbs measures for SOS model on a Cayley tree. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, 471–488 (2006).
[16] Rozikov, U. A., Shoyusupov, Sh. A.: Fertile three state HC models on Cayley tree. Theor. Math. Phys. 156, 1319-1330 (2008).
[17] Rozikov, U. A.: Gibbs measures on Cayley trees. World Sci. Publ. Singapore. (2013).
[18] Rozikov, U. A., Khakimov, R. M.: Gibbs measures for the fertile three-state hard core models on a Cayley tree. Queueing Systems. 81, 49-69 (2015).
[19] Ruelle, D.: Existence of a Phase Transition in a Continuous Classical System. Phys. Rev. Lett. 27, 1040–1041 (1971).
[20] Widom, B., Rowlinson, J. S.: New model for the study of liquid-vapor phase transition. J. Chem. Phys. 52, 1670–1684 (1970).
[21] Kissel, S., Külske, C., Rozikov, U.: Hard-core and soft-core Widom–Rowlinson models on Cayley trees. JSTAT/043204. (2019).
[22] Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, coloring and other models on trees. Random Structures and Algorithms. 31, 134–172 (2007).
[23] Rozikov, U. A., Khakimov, R. M., Makhammadaliev, M. T.: Gibbs periodic measures for a two-state HC-model on a Cayley tree. Journal of Mathematical Science, 278, 647–660 (2024).
[24] Khatamov, N. M.: Extremality of Gibbs measures for the HC-Blume–Capel model on the Cayley tree. Math. Notes. 111 (5), 768–781 (2022).
[25] Haydarov, F.H.: Kolmogorov extension theorem for non-probability measures on Cayley trees. Reviews in Mathematical Physics. 36(6), 1–12 (2024).
