发表机构
The United Arab Emirates University; V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（阿拉伯联合酋长国大学; 乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究球对称树上伊辛模型的唯一性与共存，通过分支间隙刻画，证明有界间隙等价于非零场共存，并利用返回映射分类周期情形。
AI 中文摘要
本文研究了有根球对称树上的铁磁最近邻伊辛模型，其中每个顶点的子节点数为1或q，具体由给定的分支层级序列决定。无界的一子层级序列迫使系统在任意非零均匀场和任意有限逆温度下具有唯一性。我们通过足够长的非分支段的长度来量化这一效应。相反，唯一性并不需要任何均匀速率：对于每个给定的趋于零的序列，我们构造一棵密度为1的树，其有限体积边界影响沿子序列衰减得更慢。该树保持分支数q，尽管其非零场共存区域消失。对于最终有界间隙，我们证明了与具有最大允许间隙的周期树的最优均匀比较，并得到了共存的显式场区间。因此，在此族内，有界间隙等价于在某个有限温度和某个非零场下的共存。对于最终周期间隙，吉布斯唯一性等价于全周期返回映射的不动点唯一性。负的施瓦茨导数给出完整分类：在超临界区域，映射在闭共存区间内恰有三个不动点，两个在端点，一个在外部。非零端点自动是非退化折叠，合并分支呈平方根分裂。
英文摘要
In this paper, we study the ferromagnetic nearest-neighbor Ising model on rooted spherically symmetric trees with one or \(q\) children per vertex, according to a prescribed sequence of branching levels. Unbounded runs of one-child levels force uniqueness at every nonzero homogeneous field and every finite inverse temperature. We quantify this effect through the locations of sufficiently long nonbranching stretches. Conversely, uniqueness need not entail any uniform rate: for each prescribed vanishing sequence, we construct a density-one tree whose finite-volume boundary influence decays more slowly along a subsequence. The tree retains branching number \(q\), although its nonzero-field coexistence region disappears. For eventually bounded gaps, we prove an optimal uniform comparison with the periodic tree having the largest allowed gap and obtain an explicit interval of coexistence fields. Thus bounded gaps are equivalent, within this family, to coexistence at some finite temperature and nonzero field. For eventually periodic gaps, Gibbs uniqueness is equivalent to fixed-point uniqueness of the full-period return map. A negative Schwarzian derivative yields a complete classification: in the supercritical regime the map has three fixed points inside a closed coexistence interval, two at its endpoints, and one outside. The nonzero endpoints are automatically nondegenerate folds, with square-root splitting of the merging branches.
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