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

Jiwoon Park

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arXiv · 2511.03495

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

Jiwoon Park

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Abstract

In this article, we construct and analyse a renormalisation group (RG) map for the weakly coupled $n$-component $|φ|^4$ 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.

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Jiwoon Park. 2025-11-07. A Renormalisation Group Map for Short- and Long-ranged Weakly Coupled $|φ|^4$ Models in $d \ge 4$ at and Above the Critical Point. https://arxiv.org/abs/2511.03495

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