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

We analyse and clarify the finite‐size scaling of the weakly‐coupled hierarchical ‐component model for all integers in all dimensions , for both free and periodic boundary conditions. For , we prove that for a volume of size 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 . For both boundary conditions, the average field has the same non‐Gaussian limit within a critical window of width 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 . In particular, at the infinite‐volume critical point the susceptibility scales as for periodic boundary conditions and as 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 in dimensions . For 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.