Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Critical percolation on the discrete torus in high dimensions

Arthur Blanc-Renaudie, Asaf Nachmias

TL;DR

This work establishes a precise mean-field scaling limit for critical percolation on high-dimensional discrete tori, showing that the largest clusters, properly rescaled, converge to Aldous’s Z_λ limit from critical Erdős–Rényi graphs. The authors develop a two-layer coupling framework, comprising a multiplicative component graph and a sprinkled graph, and prove a six-step contraction that forces the interaction matrix to behave essentially as rank one near criticality. Central to the argument are sharp susceptibility and moment bounds on the infinite lattice Z^d, extended to the torus, and an extended k-arm IIC construction that provides the needed fine-grained control of near-critical clusters. The results connect finite-volume torus percolation to the canonical Erdős–Rényi scaling window, demonstrating a robust mean-field universality in high dimensions and laying groundwork for future GHP-type limits of component-structure objects on finite graphs.

Abstract

We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a large enough constant for the nearest neighbor model, or any fixed $d>6$ for spread-out models. We prove that there exist constants $\mathbf{C},\mathbf{C}'$ depending only on the dimension and the spread-out parameter such that for any $λ\in \mathbb{R}$ if the edge probability is $p_c(\mathbb{Z}^d)+\mathbf{C} λn^{-d/3} + o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by $\mathbf{C}' n^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $λ-t$ at time $t\in[0,\infty)$. This canonical limit was identified by Aldous in the context of critical Erdős--Rényi graphs.

Critical percolation on the discrete torus in high dimensions

TL;DR

This work establishes a precise mean-field scaling limit for critical percolation on high-dimensional discrete tori, showing that the largest clusters, properly rescaled, converge to Aldous’s Z_λ limit from critical Erdős–Rényi graphs. The authors develop a two-layer coupling framework, comprising a multiplicative component graph and a sprinkled graph, and prove a six-step contraction that forces the interaction matrix to behave essentially as rank one near criticality. Central to the argument are sharp susceptibility and moment bounds on the infinite lattice Z^d, extended to the torus, and an extended k-arm IIC construction that provides the needed fine-grained control of near-critical clusters. The results connect finite-volume torus percolation to the canonical Erdős–Rényi scaling window, demonstrating a robust mean-field universality in high dimensions and laying groundwork for future GHP-type limits of component-structure objects on finite graphs.

Abstract

We consider percolation on the discrete torus at , the critical value for percolation on the corresponding infinite lattice , and within the scaling window around it. We assume that is a large enough constant for the nearest neighbor model, or any fixed for spread-out models. We prove that there exist constants depending only on the dimension and the spread-out parameter such that for any if the edge probability is , then the joint distribution of the largest clusters normalized by converges as to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at with drift at time . This canonical limit was identified by Aldous in the context of critical Erdős--Rényi graphs.
Paper Structure (46 sections, 45 theorems, 536 equations, 3 figures)

This paper contains 46 sections, 45 theorems, 536 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Remarks
  3. Brief background
  4. Notations and terminology
  5. About the proof
  6. The $k$-arm $r$-IIC
  7. Comparison with the recent work HypercubePaper.
  8. Subcritical building blocks
  9. Past results
  10. Sharp asymptotics of the susceptibility and expected cluster size squared
  11. Second and third moments of subcritical component sizes
  12. Subcritical estimates involving $\Delta$
  13. Organization
  14. Proof of the main theorems given \ref{['thm:main_goal']} and \ref{['sec:buildingblocks']}
  15. Proof of \ref{["thm:convergencepc'"]}
  16. ...and 31 more sections

Key Result

Theorem 1.1

Denote by $(\mathcal{C}_1, \mathcal{C}_2,\ldots)$ the connected component of percolation on the discrete torus $\mathbb{Z}_n^d$ in high dimensions with edge probability $p\in[0,1]$ sorted in non-increasing order. There exist constants $\mathbf{C}=\mathbf{C}(d,L)\in(0,\infty)$ and $\mathbf{C}'=\mathb then where $\mathbf{Z}_{\lambda}$ is the list of lengths of excursions above past minimum of an in

Figures (3)

  • Figure 4.1: The $\mathrm{Arms}(\omega_{r,R}; x_1,x_2,x_3)$ event. Dashed red lines represent open edges not in $\omega_{r,R}$ and normal blue lines are open edges in $\omega_{r,R}$. Numbers next to components of $\omega_{r,R}$ are the corresponding markings. In addition, there are no $4$ disjoint crossings.
  • Figure 5.1: The four triplets $(T,L,\rho)$ of \ref{['lem:FourSmallCases']}. In each, $L$ is colored blue, and $\rho$ is colored red.
  • Figure 5.2: Four additional triplets $(T,L,\rho)$. Again, $L$ is in blue, and $\rho$ is in red.

Theorems & Definitions (107)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 97 more