A Generalization of Cardy's and Schramm's Formulae.

Area of Science:

• Statistical physics
• Probability theory
• Conformal field theory

Background:

• Critical site percolation is a fundamental model in statistical physics.
• Understanding percolation interfaces is key to characterizing phase transitions.
• Existing formulas by Cardy and Schramm provide insights into boundary conditions.

Purpose of the Study:

• To investigate critical site percolation on the triangular lattice.
• To generalize existing formulas for percolation interfaces.
• To identify new mathematical tools for analyzing these systems.

Main Methods:

• Analysis of percolation probabilities in the scaling limit.
• Development of a novel discrete analytic observable.
• Application of conformal mapping techniques.

Main Results:

• A generalized formula for the difference in probabilities of percolation interfaces.
• The formula applies to cases where the interface does not separate two points.
• An unexpected conformal mapping was discovered in the analysis.

Conclusions:

• The study introduces a significant generalization of Cardy's and Schramm's formulas.
• A new discrete analytic observable offers fresh perspectives on percolation.
• The unexpected conformal mapping highlights deep connections within the field.