论文标题
高度功能在树上的本地化
Height function localisation on trees
论文作者
论文摘要
我们研究了两个离散高度函数的模型，即在树的顶点上的随机整数值函数的模型。首先，我们考虑随机同态模型，其中邻居必须具有一个恰好的高度差。根据定义，当地法律是统一的。我们证明，该模型的高度差异是在所有边界条件（在位置和边界高度方面）统一的界限。这意味着在系统的所有极端吉布斯测量中均匀地定位概念。对于第二个模型，我们考虑了定向树，其中每个顶点恰好有一个父母和至少两个孩子。我们将局部统一的法律视为单调的高度函数，即，父母的高度至少是子顶点的高度。我们提供了所有极端梯度吉布斯度量的完整分类，并准确地描述了该模型的局部化迁移转变。在这种情况下，典型的极端梯度吉布斯度量也是局部的。两种模型中的定位与观察到高斯自由场位于树木上的观察是一致的，这是随机行走的瞬时结果。
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.
