Annulus crossing formulae for critical planar percolation

Understand

We derive exact formulae for three basic annulus crossing events for the critical planar Bernoulli percolation in the continuum limit.

• The first is for the probability that there is an open path connecting the two boundaries of an annulus of inner radius $r$ and outer radius $R$.
• The second is for the probability that there are both open and closed paths connecting the two annulus boundaries.
• These two results were predicted by Cardy based on non-rigorous Coulomb gas arguments.