The case of equality in BK

Raphaël Cerf; Pierre Tesio

The case of equality in BK

Probability 2026-07-02 v1

Authors: Raphaël Cerf , Pierre Tesio

Abstract

We characterize the pairs of increasing events $A,B$ for which there is equality in the BK inequality. Namely, we show that $P(A\circ B)=P(A)P(B)$ if and only if all the configurations in $A\times B$ admit disjoint witnesses for $A$ and $B$ . We discuss the strengthened BK inequality, and we provide a new simplified proof of this inequality.

Keywords

BK inequality increasing events disjoint witnesses equality case percolation theory

Cite

@article{arxiv.2607.02272,
  title  = {The case of equality in BK},
  author = {Raphaël Cerf and Pierre Tesio},
  journal= {arXiv preprint arXiv:2607.02272},
  year   = {2026}
}