{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T20:03:55Z","timestamp":1771704235480,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,3]],"date-time":"2022-01-03T00:00:00Z","timestamp":1641168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we consider the \u03bb-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the \u03bb-model.<\/jats:p>","DOI":"10.3390\/a15010018","type":"journal-article","created":{"date-parts":[[2022,1,3]],"date-time":"2022-01-03T22:51:50Z","timestamp":1641250310000},"page":"18","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Extremality of Disordered Phase of \u03bb-Model on Cayley Trees"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5728-6394","authenticated-orcid":false,"given":"Farrukh","family":"Mukhamedov","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab Emirates"},{"name":"Department of Algebra and Analysis, Institute of Mathematics Named after V.I.Romanovski, 4, University Str., Tashkent 100125, Uzbekistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,3]]},"reference":[{"key":"ref_1","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, John Wiley and Sons."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0025-5564(78)90089-5","article-title":"Taxonomy with condence","volume":"40","author":"Cavender","year":"1978","journal-title":"Math. BioSci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/0092-8240(94)00051-D","article-title":"Five surprising properties of parsimoniously colored trees","volume":"57","author":"Steel","year":"1995","journal-title":"Bull. Math. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Preston, C. (1974). Gibbs States on Countable Sets, Cambridge University Press.","DOI":"10.1017\/CBO9780511897122"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1214\/aop\/1176996347","article-title":"Markov random fields on an infinite tree","volume":"3","author":"Spitzer","year":"1975","journal-title":"Ann. Probab."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Georgii, H.O. (2011). Gibbs Measures and Phase Transitions, Walter de Gruyter.","DOI":"10.1515\/9783110250329"},{"key":"ref_7","unstructured":"Sinai, Y.G. (1982). Theory of Phase Transitions: Rigorous Results, Pergamon Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Mossel, E. (2004). Survey: Information Flow on Trees. Graphs, morphisms and statistical physics. DIMACS Series Discrete Mathematics and Theoretical Computer Science 63, American Mathematical Society.","DOI":"10.1090\/dimacs\/063\/12"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/BF02108787","article-title":"Extremity of the disordered phase in the Ising model on the Bethe lattice","volume":"128","author":"Bleher","year":"1990","journal-title":"Commun. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1214\/aoap\/1019487349","article-title":"Broadcasting on trees and the Ising Model","volume":"10","author":"Evans","year":"2000","journal-title":"Ann. Appl. Probab."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF00416016","article-title":"A note on the extremality of the disordered state for the Ising model on the Bethe lattice","volume":"37","author":"Ioffe","year":"1996","journal-title":"Lett. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1103\/RevModPhys.54.235","article-title":"The Potts model","volume":"54","author":"Wu","year":"1982","journal-title":"Rev. Mod. Phys."},{"key":"ref_13","unstructured":"Baxter, R.J. (1982). Exactly Solved Models in Statistical Mechanics, Academic Press."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1214\/aoap\/1060202828","article-title":"Information ow on trees","volume":"13","author":"Mossel","year":"2003","journal-title":"Ann. Appl. Probab."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1214\/10-AOP584","article-title":"Reconstruction for the Potts model","volume":"39","author":"Sly","year":"2011","journal-title":"Ann. Probab."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1007\/BF01086840","article-title":"On pure phases of the three-state ferromagnetic Potts model on the second order Bethe lattice","volume":"85","author":"Ganikhodjaev","year":"1990","journal-title":"Theor. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1002\/rsa.20671","article-title":"Fuzzy transformations and extremaity of Gibbs measures for the Potts model on a Cayley tree","volume":"50","author":"Rozikov","year":"2017","journal-title":"Random Struct. Algorithms"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2130007","DOI":"10.1142\/S0129055X21300077","article-title":"Gibbs measures of Potts model on Cayley trees: A survey and applications","volume":"33","author":"Rozikov","year":"2021","journal-title":"Rev. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0034-4877(04)90001-8","article-title":"On factor associated with the unordered phase of \u03bb-model on a Cayley tree","volume":"53","author":"Mukhamedov","year":"2004","journal-title":"Rep. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/BF02677521","article-title":"Description of limit Gibbs measures for \u03bb-models on Bethe lattices","volume":"39","author":"Rozikov","year":"1998","journal-title":"Sib. Math. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"043204","DOI":"10.1088\/1742-5468\/ab081e","article-title":"Hard-core and soft-core Widom-Rowlinson models on Cayley trees","volume":"2019","author":"Kissel","year":"2019","journal-title":"J. Stat. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1007\/s10955-015-1279-9","article-title":"Extremality of translation-invariant phases for three-state SOS-model on the binary tree","volume":"160","author":"Rozikov","year":"2015","journal-title":"J. Stat. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rahmatullaev, M.M., and Rasulova, M.A. (2021). Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree. J. Stat. Mech., 073201.","DOI":"10.1088\/1742-5468\/ac08ff"},{"key":"ref_24","first-page":"304","article-title":"Ground states and phase transition of the \u03bb-model on the Cayley tree","volume":"193","author":"Mukhamedov","year":"2018","journal-title":"Theor. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"124184","DOI":"10.1016\/j.physa.2020.124184","article-title":"On ground states and phase Transition for \u03bb-model with the competing Potts interactions on Cayley trees","volume":"549","author":"Mukhamedov","year":"2020","journal-title":"Phys. A"},{"key":"ref_26","first-page":"1461","article-title":"Additional limit theorem for indecomposable multi-dimensional Galton-Watson processes","volume":"37","author":"Kesten","year":"1966","journal-title":"Ann. Math. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rozikov, U.A. (2013). Gibbs Measures on Cayley Trees, World Scientific.","DOI":"10.1142\/8841"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1142\/S0219025706002494","article-title":"Gibbs measures for SOS models on Cayley tree","volume":"9","author":"Rozikov","year":"2006","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. Top."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1214\/aoap\/998926994","article-title":"Reconstruction on trees: Beating the second eigenvalue","volume":"11","author":"Mossel","year":"2001","journal-title":"Ann. Appl. Probab."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1002\/rsa.20132","article-title":"Fast mixing for independent sets, coloring and other models on trees","volume":"31","author":"Martinelli","year":"2007","journal-title":"Random Struct. Algoritms"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1134\/S0040577918070103","article-title":"Extremality of the Translation-Invariant Gibbs Measures for the Potts Model on the Cayley Tree","volume":"196","author":"Rozikov","year":"2018","journal-title":"Theor. Math. Phys."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/1\/18\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:59:57Z","timestamp":1760363997000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/1\/18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,3]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["a15010018"],"URL":"https:\/\/doi.org\/10.3390\/a15010018","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,3]]}}}