  It is believed that the large-scale geometric properties of two-dimensional
critical percolation are described by a logarithmic conformal field theory, but
it has been challenging to exhibit concrete examples of logarithmic
singularities and to find an explanation and a physical interpretation, in
terms of lattice observables, for their appearance. We show that certain
percolation correlation functions receive independent contributions from a
large number of similar connectivity events happening at different scales.
Combined with scale invariance, this leads to logarithmic divergences. We study
several logarithmic correlation functions for critical percolation in the bulk
and in the presence of a boundary, including the four-point function of the
density (spin) field. Our analysis confirms previous findings, provides new
explicit calculations and explains, in terms of lattice observables, the
physical mechanism that leads to the logarithmic singularities we discover.
Although we adopt conformal field theory (CFT) terminology to present our
results, the core of our analysis relies on probabilistic arguments and recent
rigorous results on the scaling limit of critical percolation and does not
assume a priori the existence of a percolation CFT. As a consequence, our
results provide strong support for the validity of a CFT description of
critical percolation and a step in the direction of a mathematically rigorous
formulation of a logarithmic CFT of two-dimensional critical percolation.
