{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T00:45:40Z","timestamp":1773103540308,"version":"3.50.1"},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T00:00:00Z","timestamp":1694995200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study two models of <jats:italic>discrete height functions<\/jats:italic>, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the <jats:italic>random homomorphism model<\/jats:italic>, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are <jats:italic>monotone<\/jats:italic>, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.<\/jats:p>","DOI":"10.1017\/s0963548323000329","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T09:08:39Z","timestamp":1695028119000},"page":"50-64","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["Height function localisation on trees"],"prefix":"10.1017","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0017-2462","authenticated-orcid":false,"given":"Piet","family":"Lammers","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1710-8811","authenticated-orcid":false,"given":"Fabio","family":"Toninelli","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,9,18]]},"reference":[{"key":"S0963548323000329_ref28","doi-asserted-by":"publisher","DOI":"10.1214\/154957806000000050"},{"key":"S0963548323000329_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-021-01087-9"},{"key":"S0963548323000329_ref8","unstructured":"[8] Duminil-Copin, H. , Karrila, A. , Manolescu, I. and Oulamara, M. (2020) Delocalization of the height function of the six-vertex model, arXiv preprint arXiv: 2012.13750."},{"key":"S0963548323000329_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/20-AAP1647"},{"key":"S0963548323000329_ref17","doi-asserted-by":"publisher","DOI":"10.1214\/19-EJP364"},{"key":"S0963548323000329_ref18","first-page":"1128","article-title":"Dominos and the Gaussian free field","author":"Kenyon","year":"2001","journal-title":"Ann. Prob."},{"key":"S0963548323000329_ref23","doi-asserted-by":"publisher","DOI":"10.1214\/16-AOP1089"},{"key":"S0963548323000329_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01020782"},{"key":"S0963548323000329_ref12","doi-asserted-by":"publisher","DOI":"10.1515\/9783110250329"},{"key":"S0963548323000329_ref25","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548313000163"},{"key":"S0963548323000329_ref26","article-title":"Random surfaces","volume":"304","author":"Sheffield","year":"2005","journal-title":"Ast\u00e9risque"},{"key":"S0963548323000329_ref1","unstructured":"[1] Aizenman, M. , Harel, M. , Peled, R. and Shapiro, J. (2022) Depinning in integer-restricted Gaussian fields and BKT phases of two-component spin models, arXiv preprint arXiv: 2110.09498v3."},{"key":"S0963548323000329_ref4","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200008)17:1<20::AID-RSA2>3.0.CO;2-S"},{"key":"S0963548323000329_ref7","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/1012"},{"key":"S0963548323000329_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-020-03760-x"},{"key":"S0963548323000329_ref3","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1931"},{"key":"S0963548323000329_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01940327"},{"key":"S0963548323000329_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01011084"},{"key":"S0963548323000329_ref6","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v12-427"},{"key":"S0963548323000329_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01208273"},{"key":"S0963548323000329_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01608557"},{"key":"S0963548323000329_ref16","unstructured":"[16] Henning, F. and K\u00fclske, C. (2021) Existence of gradient Gibbs measures on regular trees which are not translation invariant, arXiv preprint arXiv: 2102.11899."},{"key":"S0963548323000329_ref21","unstructured":"[21] Lammers, P. and Tassy, M. (2020) Macroscopic behavior of Lipschitz random surfaces, arXiv preprint arXiv: 2004.15025."},{"key":"S0963548323000329_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-020-03920-z"},{"key":"S0963548323000329_ref22","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-015-2419-4"},{"key":"S0963548323000329_ref24","doi-asserted-by":"publisher","DOI":"10.1214\/ECP.v18-2796"},{"key":"S0963548323000329_ref27","doi-asserted-by":"publisher","DOI":"10.1214\/14-SS107"},{"key":"S0963548323000329_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-023-01202-y"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548323000329","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T09:49:28Z","timestamp":1703065768000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548323000329\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,18]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["S0963548323000329"],"URL":"https:\/\/doi.org\/10.1017\/s0963548323000329","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,18]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}