标题： 边界条件和高维层次$|\varphi|^4$模型的普遍有限尺寸标度 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： 我们分析并澄清了弱耦合分层$n$-组分$|\varphi|^4$模型在所有整数$n \ge 1$的所有维度$d\ge 4$下的有限尺寸标度，包括自由边界条件和周期性边界条件。 对于$d>4$，我们证明当体积大小为$R^{d}$且具有周期性边界条件时，无限体积临界点是一个有效的有限体积临界点，而当具有自由边界条件时，有效的临界点会减小一个阶为$R^{-2}$的量。 对于两种边界条件，平均场在有效临界点周围宽度为$R^{-d/2}$的临界窗口内具有相同的非高斯极限，在该窗口内我们计算了磁化率的普遍标度轮廓。 相反，对于两种边界条件，当高于有效临界点一个量$R^{-2}$时，平均场具有质量巨大的高斯极限。 特别是，在无限体积临界点，磁化率对于周期性边界条件按$R^{d/2}$标度，而对于自由边界条件按$R^{2}$标度。 我们识别出一种适用于自由边界条件的质量生成机制，该机制导致了这种区别，并且我们认为该机制具有更广泛的适用性，特别是适用于维度为$d \ge 4$的欧几里得（非分层）模型在$\mathbb{Z}^d$上的情况。 对于$d=4$，我们证明了具有对数修正的类似图景。 我们的分析基于Bauerschmidt、Brydges和Slade的严格重整化群方法，我们对其进行了改进和扩展。 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.