标题： 边界条件和高维（4维及以上）层次$|\varphi|^4$模型的普适有限尺寸标度特性 显示英文标题

标题： Boundary conditions and universal finite-size scaling for the hierarchical $|\varphi|^4$ model in dimensions 4 and higher

摘要： 我们分析并阐明了对于所有整数 $n \ge 1$ 在所有维度 $d\ge 4$ 下弱耦合层次 $n$-分量 $|\varphi|^4$ 模型的有限尺寸标度行为，分别考虑自由边界条件和周期性边界条件的情况。 对于 $d>4$，我们证明了当体积大小为 $R^{d}$ 且具有周期性边界条件时，无穷体积临界点是有效有限体积临界点；而对于自由边界条件，有效临界点会向小的方向偏移，偏移量为 $R^{-2}$ 的数量级。 对于两种边界条件，平均场在有效临界点附近宽度为 $R^{-d/2}$ 的临界窗口内具有相同的非高斯极限，并且在该窗口内我们计算了易感性（susceptibility）的普适缩放轮廓。相比之下，再次对于两种边界条件，当超过有效临界点一个量 $R^{-2}$ 时，平均场具有质量项的高斯极限。特别是，在无穷体积下的临界点处，周期性边界条件下的易感性按 $R^{d/2}$ 缩放，而自由边界条件下的易感性按 $R^{2}$ 缩放。我们确定了自由边界条件下的一种质量产生机制，这种机制导致了这一区别，并且我们认为它具有更广泛的适用性，特别是在维度为 $d \ge 4$ 的 $\mathbb{Z}^d$ 欧几里得（非层次结构）模型中。对于 $d=4$，我们证明了一个类似的情况，但存在对数修正。我们的分析基于 Bauerschmidt、Brydges 和 Slade 的严格重正化群方法，我们对该方法进行了改进和扩展。 显示英文摘要

摘要： We analyse and clarify the finite-size scaling of the weakly-coupled hierarchical $n$-component $|\varphi|^4$ model for all integers $n \ge 1$ in all dimensions $d\ge 4$, for both free and periodic boundary conditions. For $d>4$, we prove that for a volume of size $R^{d}$ with periodic boundary conditions the infinite-volume critical point is an effective finite-volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order $R^{-2}$. For both boundary conditions, the average field has the same non-Gaussian limit within a critical window of width $R^{-d/2}$ around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount $R^{-2}$. In particular, at the infinite-volume critical point the susceptibility scales as $R^{d/2}$ for periodic boundary conditions and as $R^{2}$ for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non-hierarchical) models on $\mathbb{Z}^d$ in dimensions $d \ge 4$. For $d=4$ we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend.