Bogachev, LV orcid.org/0000-0002-2365-2621 and Rozikov, UA (2019) On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field. Journal of Statistical Mechanics: Theory and Experiment, 2019. 073205. pp. 1-76. ISSN 1742-5468
Abstract
The paper concerns the q-state Potts model (i.e. with spin values in {1, . . . , q}) on a Cayley tree T^k of degree k ⩾ 2 (i.e. with k + 1 edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invariant SGMs and prove a novel criterion of translation invariance. Assuming a ferromagnetic nearest-neighbour spin–spin interaction, we obtain various sufficient conditions for uniqueness. For a model with constant external field, we provide in-depth analysis of uniqueness versus non-uniqueness in the subclass of completely homogeneous SGMs by identifying the phase diagrams on the ‘temperature–field’ plane for different values of the parameters q and k. In a few particular cases (e.g. q = 2 or k = 2), the maximal number of completely homogeneous SGMs in this model is shown to be 2q − 1, and we make a conjecture (supported by computer calculations) that this bound is valid for all q ⩾ 2 and k ⩾ 2.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019, IOP Publishing Ltd and SISSA Medialab srl. This is an author produced version of an article published in Journal of Statistical Mechanics: Theory and Experiment. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | classical phase transitions, exact results, phase diagrams |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 13 May 2019 14:47 |
Last Modified: | 30 Jul 2019 14:52 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1742-5468/ab270b |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:145952 |