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. 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.
I will briefly explain our probabilistic approach, which implies that the logarithmic divergence appears as a consequence of scale invariance combined with independence. Based on joint works with Federico Camia.
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