Torus scaling limits and the plateau of the critical weakly coupled $|\varphi|^4$ model in $d \ge 4$

Abstract

The $n$-component weakly coupled $|\varphi|^4$ model on the $\Z^d$ lattice ($d\ge 4$) exhibits a critical two-point correlation function with an exact polynomial decay in infinite volume, regardless of whether the interaction is short- or long-range. This paper presents a rigorous analysis of the system in both $\Z^d$ and a finite-volume torus. In a torus, we prove the existence of a plateau effect, where the correlation function undergoes a crossover from the polynomial decay to a uniform constant state. We then establish the precise scaling limit picture that provides a complementary description of this crossover. As immediate consequences, we verify the finite-size scaling limit predicted by Zinn-Justin, the finite-size scaling exponents (qoppas) suggested by Kenna and Berche and the role of the Fourier modes in finite-size scaling suggested by Flores-Sola, Berche, Kenna and Weigel. The proofs use the renormalisation group map constructed in the author's previous work.