Height function localisation on trees

IF 0.9 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Combinatorics, Probability & Computing Pub Date : 2023-09-18 DOI:10.1017/s0963548323000329

Piet Lammers, Fabio Toninelli

{"title":"Height function localisation on trees","authors":"Piet Lammers, Fabio Toninelli","doi":"10.1017/s0963548323000329","DOIUrl":null,"url":null,"abstract":"Abstract We study two models of discrete height functions , that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model , 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 monotone , 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.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability & Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s0963548323000329","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}

引用次数: 2

Abstract

Abstract We study two models of discrete height functions , that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model , 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 monotone , 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.

查看原文本刊更多论文

树的高度函数定位

摘要研究了两种离散高度函数模型，即树顶点上的随机整值函数模型。首先，我们考虑随机同态模型，其中邻居的高度差必须恰好为1。根据定义，当地法律是统一的。我们证明了该模型的高度方差是有界的，在所有边界条件下(无论是在位置上还是在边界高度上)都是均匀的。这意味着在系统的所有极值吉布斯测度上都统一地具有很强的局域化概念。对于第二个模型，我们考虑有向树，其中每个顶点恰好有一个父节点和至少两个子节点。我们考虑了单调高度函数的局部一致律，即父顶点的高度总是至少是子顶点的高度。我们提供了所有极值梯度Gibbs测度的完整分类，并准确描述了该模型的定位-离域转换。在这种情况下，典型的极值梯度吉布斯测度也是局域化的。两个模型中的局部化都与观察到的高斯自由场在树上的局部化是一致的，这是随机漫步的短暂性的直接结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Combinatorics, Probability & Computing 数学-计算机：理论方法

CiteScore

2.40

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.