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Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$

Johannes Bäumler

TL;DR

The paper proves that long-range percolation on $\mathbb{Z}^d$ with non-summable interactions ($\sum_{x\neq0} p_x=\infty$) in dimensions $d\ge 3$ retains an infinite cluster after truncating long edges, and this persists with probability tending to one as the truncation threshold grows. In the isotropic regime with sufficiently large total edge-probability, the authors obtain explicit probabilistic bounds for the infinite cluster and provide a sharp relation to the Potts model via the FK representation. The core methodology blends a second-moment analysis with carefully constructed measures on random paths and an unpredictable-path framework, enabling dimension-specific constructions and bootstrapping to control path overlaps and intersection probabilities. These results establish that non-summable long-range interactions suffice for percolation even after truncation and yield corresponding statements for long-range Potts models, highlighting the robustness of long-range connectivity in higher dimensions. The work also delineates why the same approach fails in dimension two and points to open problems, including tightening constants and extending results to $d=2$.

Abstract

Consider independent long-range percolation on $\mathbb{Z}^d$ for $d\geq 3$. Assuming that the expected degree of the origin is infinite, we show that there exists an $N\in \mathbb{N}$ such that an infinite open cluster remains after deleting all edges of length at least $N$. For the isotropic case in dimensions $d\geq 3$, we show that if the expected degree of the origin is at least $10^{400}$, then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range $q$-states Potts model.

Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$

TL;DR

The paper proves that long-range percolation on with non-summable interactions () in dimensions retains an infinite cluster after truncating long edges, and this persists with probability tending to one as the truncation threshold grows. In the isotropic regime with sufficiently large total edge-probability, the authors obtain explicit probabilistic bounds for the infinite cluster and provide a sharp relation to the Potts model via the FK representation. The core methodology blends a second-moment analysis with carefully constructed measures on random paths and an unpredictable-path framework, enabling dimension-specific constructions and bootstrapping to control path overlaps and intersection probabilities. These results establish that non-summable long-range interactions suffice for percolation even after truncation and yield corresponding statements for long-range Potts models, highlighting the robustness of long-range connectivity in higher dimensions. The work also delineates why the same approach fails in dimension two and points to open problems, including tightening constants and extending results to .

Abstract

Consider independent long-range percolation on for . Assuming that the expected degree of the origin is infinite, we show that there exists an such that an infinite open cluster remains after deleting all edges of length at least . For the isotropic case in dimensions , we show that if the expected degree of the origin is at least , then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range -states Potts model.
Paper Structure (19 sections, 14 theorems, 228 equations, 1 figure)

This paper contains 19 sections, 14 theorems, 228 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Open problems
  5. Dispersion of random paths
  6. Sums of random variables
  7. Unpredictable paths
  8. Proofs of the Theorems
  9. Proof of \ref{['eq:main high prob']} and \ref{['eq:isotrop finite cluster low prob']}
  10. The general setup
  11. Proof of Theorem \ref{['theo:main']}
  12. Proof of Theorem \ref{['theo:isotropic']}: Measures on random paths
  13. Random paths in dimensions $d \geq 4$
  14. Random paths in dimension $d = 3$
  15. The bootstrapping lemmas
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $d\geq 3$, and assume that $\left(p_x\right)_{x\in \mathbb{Z}^d}$ satisfies conditions eq:non-summable through eq:irreduc. Then there exists $n\in \N$ such that Furthermore,

Figures (1)

  • Figure 1: Two self-avoiding paths starting at the origin. The edges which are traversed by both paths (f, g, and h) are drawn in purple. Both paths traverse the edges f and g in the same direction. Contrary to that, the paths traverse the edge h in opposite directions.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • proof
  • Conjecture 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 28 more