# Upper bounds on the percolation correlation length

## TL;DR

This paper establishes an exponential upper bound on the correlation length in near-critical Bernoulli percolation on integer lattices, advancing understanding of phase transition behavior.

## Contribution

It provides a new quantitative bound on the correlation length near criticality, improving previous estimates and supporting the conjecture about the absence of infinite clusters at criticality.

## Key findings

- Correlation length bounded by exp(C/|p-p_c|^2)
- Supports the conjecture of no infinite cluster at criticality
- Advances quantitative understanding of near-critical percolation

## Abstract

We study the size of the near-critical window for Bernoulli percolation on $\mathbb Z^d$. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by $\exp(C/|p-p_c|^2)$. Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on $\mathbb Z^d$ for every $d\ge2$.