Logarithmic correlation functions in 2D critical percolation

TL;DR

This paper demonstrates how certain correlation functions in 2D critical percolation exhibit logarithmic divergences due to scale-invariant connectivity events, supporting the conjecture of a logarithmic conformal field theory description.

Contribution

It provides explicit calculations and a physical interpretation of logarithmic singularities in critical percolation without assuming a pre-existing conformal field theory framework.

Findings

Logarithmic divergences arise from scale-invariant connectivity events.

Explicit correlation functions confirm the presence of logarithmic singularities.

Results support the existence of a logarithmic CFT description for 2D critical percolation.

Abstract

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.