A Renormalisation Group Map for Short- and Long-ranged Weakly Coupled $|\varphi|^4$ Models in $d \ge 4$ at and Above the Critical Point

TL;DR

This paper develops a renormalisation group map for weakly coupled $||^4$ models in dimensions $d \,\geq\, 4$, considering both short- and long-range interactions, to analyze correlation decay and finite-volume effects.

Contribution

It extends existing RG maps to include long-range interactions and finite-volume effects, enabling precise correlation decay analysis in higher dimensions.

Findings

Established decay rates of correlation functions.

Provided evidence for a plateau in finite-volume systems.

Refined RG analysis for models with both short- and long-range interactions.

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.