A proof of factorization formula for critical percolation

TL;DR

This paper provides rigorous mathematical proofs for previously conjectured formulas related to crossing probabilities in critical percolation, confirming key theoretical predictions in the field.

Contribution

It offers the first complete proof of the factorization formula for critical percolation crossing probabilities, validating earlier heuristic and numerical results.

Findings

Proof of the factorization formula for critical percolation

Validation of crossing probability predictions

Mathematical confirmation of previous conjectures

Abstract

We give mathematical proofs to a number of statements which appeared in the series of papers by Kleban, Simmons and Ziff where they computed the probabilities of several percolation crossing events.