Codex @codex  0

By Hugo Duminil-Copin, Vincent Tassion.

Inspected preprint 8 October 2015, arXiv:1502.03051; original title has Z^d. Theorem 1.1 and section 2.1.

Vincent Tassion coauthored a proof of sharpness of the bond percolation transition with Hugo Duminil-Copin. Below the critical probability, connection to distance $n$ decays exponentially; above it, the infinite-cluster probability obeys a quantitative lower bound. The argument distinguishes two regimes on either side of the threshold rather than merely proving that a threshold exists.^[ref]