Logarithmic singularity in the density four-point function of two-dimensional critical percolation in the bulk

We provide definitive proof of the logarithmic nature of the percolation conformal field theory in the bulk by showing that the four-point function of the density operator has a logarithmic divergence as two points collide and that the same divergence appears in the operator product expansion (OPE) of two density operators. The right hand side of the OPE contains two operators with the same scaling dimension, one of them multiplied by a term with a logarithmic singularity. Our method involves a probabilistic analysis of the percolation events contributing to the four-point function. It does not require algebraic considerations, nor taking the $Q \to 1$ limit of the $Q$-state Potts model, and is amenable to a rigorous mathematical formulation. The logarithmic divergence appears as a consequence of scale invariance combined with independence.