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Sharp bounds on the half-space two-point function for high-dimensional Bernoulli percolation

Romain Panis, Bruno Schapira

TL;DR

This work settles sharp up-to-constant bounds for the critical two-point function in the half-space $\mathbb{H}$ for high-dimensional Bernoulli percolation with $d>6$, under the standard full-space decay $\tau(x,y)\asymp 1/(1+|x-y|^{d-2})$. The authors combine a BK-based path-decomposition with two novel ingredients: a reversed BK inequality that handles boundary-restricted contributions, and a capacity-driven lower bound expressed via $(d-4)$-capacity. They introduce a regularity framework (regular points, line good points, extended cluster) inspired by Kozma–Nachmias and capacity methods, enabling precise control of boundary effects and pioneer-like connections from the bulk to the boundary. The main result provides an explicit half-space decay $\tau_{\mathbb{H}}(x,y) \asymp \frac{(1+r_{x,y})(1+r_{y,x})}{1+|x-y|^{d}}$, resolving a conjecture of Hutchcroft–Michta–Slade and advancing understanding of boundary phenomena in mean-field percolation with potential implications for pioneers and capacity-based analyses.

Abstract

We consider Bernoulli percolation on $\mathbb Z^d$ with $d>6$. We prove an up-to-constant estimate for the critical two-point function restricted to a half-space. This completes previous results of Chatterjee and Hanson (Commun. Pure Appl. Math., 2021), and Chatterjee, Hanson, and Sosoe (Commun. Math. Phys., 2023), and solves a question asked by Hutchcroft, Michta, and Slade (Ann. Probab., 2023).

Sharp bounds on the half-space two-point function for high-dimensional Bernoulli percolation

TL;DR

This work settles sharp up-to-constant bounds for the critical two-point function in the half-space for high-dimensional Bernoulli percolation with , under the standard full-space decay . The authors combine a BK-based path-decomposition with two novel ingredients: a reversed BK inequality that handles boundary-restricted contributions, and a capacity-driven lower bound expressed via -capacity. They introduce a regularity framework (regular points, line good points, extended cluster) inspired by Kozma–Nachmias and capacity methods, enabling precise control of boundary effects and pioneer-like connections from the bulk to the boundary. The main result provides an explicit half-space decay , resolving a conjecture of Hutchcroft–Michta–Slade and advancing understanding of boundary phenomena in mean-field percolation with potential implications for pioneers and capacity-based analyses.

Abstract

We consider Bernoulli percolation on with . We prove an up-to-constant estimate for the critical two-point function restricted to a half-space. This completes previous results of Chatterjee and Hanson (Commun. Pure Appl. Math., 2021), and Chatterjee, Hanson, and Sosoe (Commun. Math. Phys., 2023), and solves a question asked by Hutchcroft, Michta, and Slade (Ann. Probab., 2023).

Paper Structure

This paper contains 11 sections, 12 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations.
  3. The van den Berg--Kesten inequality.
  4. Proof of Theorem \ref{['thm:main']}
  5. Proof of Proposition \ref{['prop.phik']}
  6. Proof of Proposition \ref{['prop:reversedBK']}
  7. A general lower bound for the connection probability of two finite sets
  8. Regular points, line good points, extended cluster
  9. Conclusion
  10. Proof of Lemma \ref{['Prop:regpoints']}
  11. Acknowledgements.

Key Result

Proposition 1.1

Let $d>6$ and assume that eq: 2pt full space estimate holds. Then, for every $K\geq 1$, there exist $c,C>0$ such that the following holds:

Figures (2)

  • Figure 1: An illustration of the decomposition used to obtain \ref{['upperboundbyBK']} (in the nearest-neighbour case). The black bold path represents an open self-avoiding path from $x$ to $y$. Depending on the values of $x_1$ and $y_1$, the boxes $B_n(x)$ and $B_n(y)$ may "touch" the boundary of $\mathbb H$. The reversed inequality of Proposition \ref{['prop:reversedBK']} decomposes paths from $x$ to $y$ similarly, except that there is an additional restriction to vertices $u$ and $v$ satisfying $u_1,v_1\geq \varepsilon n$.
  • Figure 2: An illustration of the diagrams underlying the proof of \ref{["eq.aa'bb'"]}. The black bold paths are open self-avoiding paths. If the event $\{a\xleftrightarrow{\mathbb H\:}b, a'\xleftrightarrow{\mathbb H\:}b'\}$ occurs, then one of the situations must occur (for some $w,w'\in \mathbb H$). Each diagram corresponds to a term on the right-hand side of \ref{["eq.aa'bb'"]}.

Theorems & Definitions (26)

  • Proposition 1.1: ChatterjeeHanson
  • Proposition 1.2: ChatterjeeHansonSosoe2023subcritical
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • proof : Proof of Corollary \ref{['coro: pionneers']}
  • Proposition 2.1
  • Proposition 2.2
  • ...and 16 more