Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuity of the critical value and a shape theorem for long-range percolation

Johannes Bäumler

TL;DR

The paper develops a comprehensive continuity framework for long-range percolation on $\mathbb{Z}^d$ by analyzing kernel perturbations and short-edge variations. Using a Grimmett–Marstrand approach and the notion of kernel resilience, it proves that the critical parameter $\beta_c$ depends continuously on the kernel under both $L^1$-convergence and finite-range truncation, and that the percolation probability is continuous away from criticality. It then establishes a shape theorem for the long-range percolation metric in the strong-decay regime, along with transience of the infinite cluster in $d\ge 3$, and shows the existence of large clusters in the supercritical regime. The work also develops a strict inequality for critical points to extend continuity results to models with perturbations of short-edge probabilities, and it highlights the breakdown of locality in one dimension, delineating the boundaries of locality phenomena in long-range systems.

Abstract

We show that for long-range percolation with polynomially decaying connection probabilities in dimensions $d\geq 2$, the critical value depends continuously on the precise specifications of the model. We use this result to prove a shape theorem for super-critical long-range percolation in the strong decay regime and to show transience of the infinite supercritical long-range percolation cluster in dimensions $d\geq 3$.

Continuity of the critical value and a shape theorem for long-range percolation

TL;DR

The paper develops a comprehensive continuity framework for long-range percolation on by analyzing kernel perturbations and short-edge variations. Using a Grimmett–Marstrand approach and the notion of kernel resilience, it proves that the critical parameter depends continuously on the kernel under both -convergence and finite-range truncation, and that the percolation probability is continuous away from criticality. It then establishes a shape theorem for the long-range percolation metric in the strong-decay regime, along with transience of the infinite cluster in , and shows the existence of large clusters in the supercritical regime. The work also develops a strict inequality for critical points to extend continuity results to models with perturbations of short-edge probabilities, and it highlights the breakdown of locality in one dimension, delineating the boundaries of locality phenomena in long-range systems.

Abstract

We show that for long-range percolation with polynomially decaying connection probabilities in dimensions , the critical value depends continuously on the precise specifications of the model. We use this result to prove a shape theorem for super-critical long-range percolation in the strong decay regime and to show transience of the infinite supercritical long-range percolation cluster in dimensions .
Paper Structure (17 sections, 24 theorems, 197 equations, 4 figures)

This paper contains 17 sections, 24 theorems, 197 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Varying short edges only
  4. Notation
  5. The proof of Theorem \ref{['theo:main']}
  6. Applications of Theorem \ref{['theo:main']}
  7. Locality of long-range percolation
  8. Continuity of the percolation probability outside criticality
  9. Existence of large clusters
  10. Transience of random walks
  11. A shape theorem for the long-range percolation metric
  12. Proof of Theorem \ref{['theo:chemical dist']}
  13. Proofs of the Propositions
  14. Varying short edges only
  15. Strict inequality of critical points
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d\geq 2$ and let $J:\mathbb{Z}^d \to \left[0,\infty\right)$ be an irreducible and symmetric kernel such that $J(x)=\mathcal{O}(\|x\|^{-2d})$. Let $\beta > \beta_c\left(J\right)$. Then there exists $N \in \N$ so that the kernel $\tilde{J}$ defined by satisfies $\mathbb{P}_{\beta, \tilde{J}}\left(|K_\mathbf{0}|=\infty\right)>0$.

Figures (4)

  • Figure 1: An illustration of the statement of Lemma \ref{['lem3']}: The inner blue square $(R)$ is connected by an open path (the black edges) to an open $m$-pad (the outer blue square) in $S_n^{(1+\delta)n} \cap B_{\delta n}(y)$ (the green hatched area).
  • Figure 2: Connecting $m$-pads (the blue boxes) in the proof of Lemma \ref{['lem4']}. We first find a path from $B_m(\mathbf{0})$ (the blue box on the left side) to $A = B_m(z)$ (the middle blue box). From $A$, we find a path to an $m$-pad $\tilde{A} \subset F_m^{\delta}(ne_1)$ (in the picture, $\tilde{A}$ is the blue box on the right side). Concatenating these two paths gives a path from $B_m(\mathbf{0})$ to $F_m^{\delta}(ne_1)$.
  • Figure 3: An illustration of the statement of Corollary \ref{['coro:conn']}: $B_m(\mathbf{0})$ (the left blue square) is connected by an open path (the black edges) to an open $m$-pad (right blue square) in the target area (green hatched). The path does not leave the big rectangle.
  • Figure 4: An illustration of the statement of Corollary \ref{['coro:steer2']} in dimension $d=2$: For every set $B_m(u) \subset B_n(\mathbf{0})$ (the blue square on the left side as a subset of the orange hatched area) there exists with high probability a path (the black edges) to an open $m$-pad (the blue square on the right) in the target area $T_1$ (the green hatched area). This path does not use edges outside the big $(14n+1) \times (6n+1)$ rectangle, which is $M_1$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Proposition 1.10
  • ...and 44 more