Geodesic
Fixed points of an infinite-dimensional operator related to Gibbs measures
Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 329-344 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set $\mathbb N$ of spin values on a Cayley tree. This operator is defined by a countable set of parameters $\lambda_i>0$, $a_{ij}\in\{0,1\}$, $i,j\in\mathbb N$. We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model.
Keywords: fixed point, Cayley tree, Gibbs measure, HC model. 
@article{TMF_2023_214_2_a11,
     author = {U. R. Olimov and U. A. Rozikov},
     title = {Fixed points of an infinite-dimensional operator related to {Gibbs} measures},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {329--344},
     year = {2023},
     volume = {214},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/}
}
TY  - JOUR
AU  - U. R. Olimov
AU  - U. A. Rozikov
TI  - Fixed points of an infinite-dimensional operator related to Gibbs measures
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 329
EP  - 344
VL  - 214
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/
LA  - ru
ID  - TMF_2023_214_2_a11
ER  - 
%0 Journal Article
%A U. R. Olimov
%A U. A. Rozikov
%T Fixed points of an infinite-dimensional operator related to Gibbs measures
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 329-344
%V 214
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/
%G ru
%F TMF_2023_214_2_a11
U. R. Olimov; U. A. Rozikov. Fixed points of an infinite-dimensional operator related to Gibbs measures. Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 329-344. http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/

[1] S. Zachary, “Countable state space Markov random fields and Markov chains on trees”, Ann. Probab., 11:4 (1983), 894–903 | DOI | MR

[2] M. Biskup, R. Kotecký, “Phase coexistence of gradient Gibbs states”, Probab. Theory Related Fields, 139:1–2 (2007), 1–39 | DOI | MR

[3] R. Bissacot, E. O. Endo, A. C. D. van Enter, “Stability of the phase transition of critical-field Ising model on Cayley trees under inhomogeneous external fields”, Stoch. Process. Appl., 127:12 (2017), 4126–4138 | DOI | MR

[4] S. Friedli, Y. Velenik, Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction, Cambridge Univ. Press, Cambridge, 2018 | MR

[5] T. Funaki, H. Spohn, “Motion by mean curvature from the Ginzburg–Landau interface model”, Comm. Math. Phys., 185:1 (1997), 1–36 | DOI | MR

[6] N. N. Ganikhodzhaev, U. A. Rozikov, N. M. Khatamov, “Mery Gibbsa dlya modeli HC–Blyuma–Kapelya so schetnym chislom sostoyanii na dereve Keli”, TMF, 211:3 (2022), 491–501 | DOI | DOI | MR

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

[8] F. H. Haydarov, U. A. Rozikov, “Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws”, Rep. Math. Phys., 90:1 (2022), 81–101, arXiv: 2110.10078 | DOI | MR

[9] U. A. Rozikov, F. H. Haydarov, “A HC model with countable set of spin values: Uncountable set of Gibbs measures”, Rev. Math. Phys., 2022, 17 pp., Online ready, , arXiv: 2206.06333 | DOI

[10] F. Henning, C. Külske, A. Le Ny, U. A. Rozikov, “Gradient Gibbs measures for the SOS model with countable values on a Cayley tree”, Electron. J. Probab., 24 (2019), 104, 23 pp. | DOI | MR

[11] F. Henning, C. Külske, Existence of gradient Gibbs measures on regular trees which are not translation invariant, arXiv: 2102.11899

[12] F. Henning, C. Külske, “Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees”, Ann. Appl. Probab., 31:5 (2021), 2284–2310 | DOI | MR

[13] F. Henning, Gibbs measures and gradient Gibbs measures on regular trees, PhD thesis, Ruhr-Universität, Bochum, 2021

[14] C. Külske, P. Schriever, “Gradient Gibbs measures and fuzzy transformations on trees”, Markov Process. Relat. Fields, 23 (2017), 553–590, arXiv: 1609.00159

[15] C. Külske, Stochastic Processes on Trees, Ruhr-Universität, Bochum, 2017 https://www.ruhr-uni-bochum.de/imperia/md/content/mathematik/kuelske/stoch-procs-on-trees.pdf

[16] U. A. Rozikov, “Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees”, J. Stat. Phys., 188:3 (2022), 26, 16 pp., arXiv: 2203.11446 | DOI | MR

[17] S. Sheffield, Random surfaces: Large deviations principles and gradient Gibbs measure classifications, PhD thesis, Stanford University, 2003 | MR

[18] Y. Velenik, “Localization and delocalization of random interfaces”, Probab. Surv., 3 (2006), 112–169 | DOI | MR

[19] 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. | MR

[20] C. Külske, U. A. Rozikov, 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, arXiv: 1310.6220 | DOI | MR

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

[22] U. A. Rozikov, Gibbs measures in Biology and Physics: The Potts Model, World Sci., Singapore, 2022 | DOI