Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jiwoon Park

TL;DR

The paper delivers a rigorous finite-size scaling framework for the high-dimensional $| abla\varphi|^4$ model with short- and long-range interactions by constructing a stable RG flow around a carefully tuned critical point and analyzing both local and global fluctuations. It proves a torus scaling limit featuring a non-Gaussian plateau for the 0-mode and Gaussian-type fluctuations for nonzero Fourier modes, tying these results to precise scaling exponents and Fourier-mode decomposition. The work confirms finite-size scaling predictions at and above the upper critical dimension, including the role of Fourier modes and the qoppa exponents, and provides a robust RG-based mechanism for understanding plateau phenomena in finite volumes. The combination of a detailed covariance decomposition, a stabilized RG map, and a rigorous treatment of 0-mode fluctuations yields a comprehensive picture of FSS above $d_{c,u}$ with both white-noise and Gaussian scaling limits, and highlights several open directions for extending these results to broader parameter regimes and boundary conditions.

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.

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

TL;DR

The paper delivers a rigorous finite-size scaling framework for the high-dimensional model with short- and long-range interactions by constructing a stable RG flow around a carefully tuned critical point and analyzing both local and global fluctuations. It proves a torus scaling limit featuring a non-Gaussian plateau for the 0-mode and Gaussian-type fluctuations for nonzero Fourier modes, tying these results to precise scaling exponents and Fourier-mode decomposition. The work confirms finite-size scaling predictions at and above the upper critical dimension, including the role of Fourier modes and the qoppa exponents, and provides a robust RG-based mechanism for understanding plateau phenomena in finite volumes. The combination of a detailed covariance decomposition, a stabilized RG map, and a rigorous treatment of 0-mode fluctuations yields a comprehensive picture of FSS above with both white-noise and Gaussian scaling limits, and highlights several open directions for extending these results to broader parameter regimes and boundary conditions.

Abstract

The -component weakly coupled model on the lattice () 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 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.

Paper Structure

This paper contains 65 sections, 70 theorems, 390 equations.

Table of Contents

  1. Introduction and main results
  2. Definition of the model
  3. Limiting scales and measures
  4. Two-point function and plateau
  5. Near-critical point limits
  6. Torus scaling limit at the critical point
  7. Role of Fourier modes
  8. $\hbox{{\fontencoding{LGR}\textqoppa}}$ and $\widehat{{\hbox{\footnotesize{\fontencoding{LGR}\textqoppa}}}}$ exponents
  9. Open problems and future directions
  10. Outline of the method
  11. Notation
  12. Gaussian integrals and effective potential functions
  13. Fractional Laplacian
  14. Modifications to the covariance
  15. Covariance decomposition
  16. ...and 50 more sections

Key Result

lemma 1

Let $Y$ be a $\R^n$-valued random variable given by eq:Ydistn. Then $Y \one =^d \operatorname{\bf NG}$.

Theorems & Definitions (151)

  • definition 1
  • lemma 1
  • proof
  • theorem 1
  • remark 1
  • theorem 2
  • theorem 3
  • corollary 1
  • proof
  • remark 2
  • ...and 141 more