Convergence of $k$-point functions in high dimensional percolation

TL;DR

This paper proves that in high-dimensional critical Bernoulli percolation, the probability of multiple points being in the same cluster converges to a constant after rescaling, confirming a longstanding conjecture.

Contribution

It establishes the convergence of multi-point connection probabilities in high-dimensional percolation, confirming a conjecture by Aizenman and Newman.

Findings

Probability of points in the same cluster converges after rescaling.

Explicit constant for the limit probability is identified.

Confirms a conjecture on high-dimensional percolation behavior.

Abstract

Consider critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large; let $y_0, \dots, y_{k-1}$ be $k$ distinct points in $\mathbb{R}^d$. We prove that the probability that $\{\lfloor n y_i\rfloor\}_{i=0}^{k-1}$ all lie in the same open cluster, rescaled by an appropriate power of $n$, converges as $n \to \infty$ to an explicit constant. This confirms a conjecture of Aizenman and Newman.