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Robust construction of the incipient infinite cluster in high dimensional critical percolation

Shirshendu Chatterjee, Pranav Chinmay, Jack Hanson, Philippe Sosoe

TL;DR

This work constructs the incipient infinite cluster (IIC) for high-dimensional Bernoulli percolation without relying on lace-expansion techniques, by exploiting two-point function asymptotics and a Kesten-style Markov-chain framework. It provides two robust routes to the IIC: conditioning on $0$ being in an infinite cluster for $p>p_c$ and letting $p\searrow p_c$, or conditioning on a connection to a distant set $V_n$ at criticality and letting $n\to\infty$, with the limiting measure $\nu$ shown to be independent of the specific conditioning family. The construction rests on replacing RSW/division-into-circuits arguments with gluing of large spanning clusters and a regularity theory (KN/CH), together with a hierarchical annulus decomposition and a matrix-product transition structure that is analyzed via Hopf’s lemma. The results apply in all dimensions where the high-dimensional two-point function asymptotics are known (e.g., $d>6$ for spread-out models and $d\ge 11$ for nearest-neighbor models), yielding a lace-expansion-free route to the IIC and paving the way for scaling-limit analyses of near-critical percolation clusters in high dimensions.

Abstract

We give a new construction of the incipient infinite cluster (IIC) associated with high-dimensional percolation in a broad setting and under minimal assumptions. Our arguments differ substantially from earlier constructions of the IIC; we do not directly use the machinery of the lace expansion or similar diagrammatic expansions. We show that the IIC may be constructed by conditioning on the cluster of a vertex being infinite in the supercritical regime $p > p_c$ and then taking $p \searrow p_c$. Furthermore, at criticality, we show that the IIC may be constructed by conditioning on a connection to an arbitrary distant set $V$, generalizing previous constructions where one conditions on a connection to a single distant vertex or the boundary of a large box. The input to our proof are the asymptotics for the two-point function obtained by Hara, van der Hofstad, and Slade. Our construction thus applies in all dimensions for which those asymptotics are known, rather than an unspecified high dimension considered in previous works. The results in this paper will be instrumental in upcoming work related to structural properties and scaling limits of various objects involving high-dimensional percolation clusters at and near criticality.

Robust construction of the incipient infinite cluster in high dimensional critical percolation

TL;DR

This work constructs the incipient infinite cluster (IIC) for high-dimensional Bernoulli percolation without relying on lace-expansion techniques, by exploiting two-point function asymptotics and a Kesten-style Markov-chain framework. It provides two robust routes to the IIC: conditioning on being in an infinite cluster for and letting , or conditioning on a connection to a distant set at criticality and letting , with the limiting measure shown to be independent of the specific conditioning family. The construction rests on replacing RSW/division-into-circuits arguments with gluing of large spanning clusters and a regularity theory (KN/CH), together with a hierarchical annulus decomposition and a matrix-product transition structure that is analyzed via Hopf’s lemma. The results apply in all dimensions where the high-dimensional two-point function asymptotics are known (e.g., for spread-out models and for nearest-neighbor models), yielding a lace-expansion-free route to the IIC and paving the way for scaling-limit analyses of near-critical percolation clusters in high dimensions.

Abstract

We give a new construction of the incipient infinite cluster (IIC) associated with high-dimensional percolation in a broad setting and under minimal assumptions. Our arguments differ substantially from earlier constructions of the IIC; we do not directly use the machinery of the lace expansion or similar diagrammatic expansions. We show that the IIC may be constructed by conditioning on the cluster of a vertex being infinite in the supercritical regime and then taking . Furthermore, at criticality, we show that the IIC may be constructed by conditioning on a connection to an arbitrary distant set , generalizing previous constructions where one conditions on a connection to a single distant vertex or the boundary of a large box. The input to our proof are the asymptotics for the two-point function obtained by Hara, van der Hofstad, and Slade. Our construction thus applies in all dimensions for which those asymptotics are known, rather than an unspecified high dimension considered in previous works. The results in this paper will be instrumental in upcoming work related to structural properties and scaling limits of various objects involving high-dimensional percolation clusters at and near criticality.

Paper Structure

This paper contains 15 sections, 15 theorems, 180 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous results and motivation
  4. Future and ongoing work.
  5. Outline of the proof
  6. Preliminaries and definitions
  7. Notation and useful results
  8. Initial preparation
  9. Restricting to finite boxes
  10. Regularity
  11. Gluing constructions and lemmas
  12. Structural properties of large open clusters
  13. Decomposing the open arm from $0$
  14. Initial decomposition
  15. Kesten's Markov Chain argument

Key Result

Theorem 1.1

Let $\mathbb{P}$ denote the product Bernoulli measure on $\{0,1\}^{\mathsf{E}(\mathbb{Z}^d)}$, with common parameter $p=p_c$, the critical density. Let also $d>6$ be such that the estimates 2pt hold. (For example, $d\ge11$ suffices FHFH2.) Suppose that, for each $n$, the sets $V_n, D_n$ are subsets exists, where the limit is in the sense of weak convergence.

Figures (3)

  • Figure 1: The event $\mathcal{E}_2(x_i', x_j') \cap \{x_i \,{\buildrel \mathbb{Z}^d \setminus \mathcal{D} \over \longleftrightarrow}\, x_i'\}$.
  • Figure 2: The event $\mathcal{E}_2(x_i', x_j') \cap \{x_j \,{\buildrel \mathbb{Z}^d \setminus \mathcal{C} \over \longleftrightarrow}\, x_i'\}$.
  • Figure 3: The event $\mathcal{E}_2(x_i', x_j') \cap \{x_i \leftrightarrow x_j\} \setminus \{x_i \leftrightarrow x_i'\}$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:glue']}, item \ref{['it:twoclust']}
  • Proposition 3.3
  • ...and 19 more