# A Generalization of Cardy’s and Schramm’s Formulae

## TL;DR

This paper extends two key formulas in percolation theory using a new mathematical approach on a triangular lattice.

## Contribution

The paper introduces a new discrete analytic observable and an unexpected conformal mapping to generalize existing percolation formulas.

## Key findings

- The difference in interface probabilities is calculated in the scaling limit.
- The approach generalizes both Cardy’s and Schramm’s formulae.
- A new discrete analytic observable was found to be crucial for the generalization.

## Abstract

We study critical site percolation on the triangular lattice. We find the difference of the probabilities of having a percolation interface to the right and to the left of two given points (such that the union of the triangles intersecting the interface does not separate the points) in the scaling limit. This generalizes both Cardy’s and Schramm’s formulae. The generalization involves a new interesting discrete analytic observable and an unexpected conformal mapping.