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Boundary conditions and the two-point function plateau for the hierarchical $|\varphi|^4$ model in dimensions 4 and higher

Jiwoon Park, Gordon Slade

TL;DR

This paper proves that in dimensions $d\ge 4$ the finite-volume hierarchical $|\

Abstract

We obtain precise plateau estimates for the two-point function of the finite-volume weakly-coupled hierarchical $|\varphi|^4$ model in dimensions $d \ge 4$, for both free and periodic boundary conditions, and for any number $n \ge 1$ of components of the field $\varphi$. We prove that, within a critical window around their respective effective critical points, the two-point functions for both free and periodic boundary conditions have a plateau, in the sense that they decay as $|x|^{-(d-2)}$ until reaching a constant plateau value of order $V^{-1/2}$ (with a logarithmic correction for $d=4$), where $V$ is size of the finite volume. The two critical windows for free and periodic boundary conditions do not overlap. The dependence of the plateau height on the location within the critical window is governed by an explicit $n$-dependent universal profile which is independent of the dimension. The proof is based on a rigorous renormalisation group method and extends the method used by Michta, Park and Slade (arXiv:2306.00896) to study the finite-volume susceptibility and related quantities. Our results lead to precise conjectures concerning Euclidean (non-hierarchical) models of spin systems and self-avoiding walk in dimensions $d \ge 4$.

Boundary conditions and the two-point function plateau for the hierarchical $|\varphi|^4$ model in dimensions 4 and higher

TL;DR

This paper proves that in dimensions the finite-volume hierarchical $|\

Abstract

We obtain precise plateau estimates for the two-point function of the finite-volume weakly-coupled hierarchical model in dimensions , for both free and periodic boundary conditions, and for any number of components of the field . We prove that, within a critical window around their respective effective critical points, the two-point functions for both free and periodic boundary conditions have a plateau, in the sense that they decay as until reaching a constant plateau value of order (with a logarithmic correction for ), where is size of the finite volume. The two critical windows for free and periodic boundary conditions do not overlap. The dependence of the plateau height on the location within the critical window is governed by an explicit -dependent universal profile which is independent of the dimension. The proof is based on a rigorous renormalisation group method and extends the method used by Michta, Park and Slade (arXiv:2306.00896) to study the finite-volume susceptibility and related quantities. Our results lead to precise conjectures concerning Euclidean (non-hierarchical) models of spin systems and self-avoiding walk in dimensions .
Paper Structure (49 sections, 43 theorems, 320 equations)

This paper contains 49 sections, 43 theorems, 320 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. The model
  4. The hierarchial group
  5. The hierarchial Laplacian
  6. The hierarchical Green function
  7. The hierarchical $|\varphi|^4$ model
  8. The critical point
  9. The effective critical points and critical windows
  10. Main results
  11. The plateau in the non-Gaussian regime
  12. Absence of plateau in the Gaussian regime
  13. Open problems
  14. Euclidean (non-hierarchial) models
  15. Self-avoiding walk
  16. ...and 34 more sections

Key Result

theorem 1

Let $d\ge 4$, let $n \in \N$, let $L$ be sufficiently large, and let $g>0$ be sufficiently small (depending on $L$). There exist a critical value $\nu_c \in \R$ and a constant $A_d>0$ (both depending on $d,g,n,L$) such that the infinite-volume limit of the susceptibility exists for $\nu=\nu_c+\varep

Theorems & Definitions (83)

  • theorem 1
  • definition 1
  • theorem 2
  • corollary 1
  • corollary 2
  • proof
  • theorem 3
  • corollary 3
  • corollary 4
  • proof
  • ...and 73 more