A Renormalisation Group Map for Short- and Long-ranged Weakly Coupled $|\varphi|^4$ Models in $d \ge 4$ at and Above the Critical Point
Jiwoon Park
TL;DR
This work provides a rigorous construction of a renormalisation group map for weakly coupled $n$-component $||^4$ models on $d \ge 4$, incorporating both short-range and long-range interactions. By representing the problem in a polymer-expansion framework and employing an extended norm, the authors establish contraction and stability estimates across scales, enabling precise control of the RG flow and the local-infinite-volume limit. The results pave the way to exact correlation-decay rates and illuminate finite-volume plateau phenomena in finite systems with periodic boundaries, bridging rigorous RG methods with finite-size scaling predictions for higher-dimensional and long-range spin models. The approach extends the Bauerschmidt–Brydges–Slade program to new regimes, providing a unified treatment of short-range and long-range weakly coupled $||^4$ theories and opening avenues for rigorous analysis of scaling limits and observables in $O(n)$ models at and above the critical point.
Abstract
In this article, we construct and analyse a renormalisation group (RG) map for the weakly coupled $n$-component $|\varphi|^4$ model under periodic boundary conditions in dimension $d \ge 4$. Both short-range and long-range interactions with upper critical dimension four are considered. This extends and refines the RG map constructed by Bauerschmidt, Brydges and Slade for the short-range model at $d=4$. This extension opens the door to establishing the exact decay rate of correlation functions of all of the models discussed. Furthermore, incorporating a large-field decay estimate and comparing with the finite-size scaling results of Michta, Park, and Slade, our analysis provides strong evidence for the emergence of a plateau in systems of finite volume with periodic boundary conditions.