Logarithmic observables in critical percolation

TL;DR

This paper demonstrates that certain observables in critical percolation exhibit logarithmic scaling, confirming predictions of logarithmic conformal field theory through analytical derivations and Monte Carlo simulations.

Contribution

It identifies specific observables in critical percolation with logarithmic two-point functions and computes the universal logarithmic prefactor exactly.

Findings

Logarithmic two-point functions are observed at Q=1 in the Potts model.

The universal logarithmic prefactor is computed exactly in 2D.

Monte Carlo simulations confirm the theoretical predictions.

Abstract

Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q = 1 limit of the Q-state Potts model, and analyzing the underlying S_Q symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q = 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d=2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by extensive Monte-Carlo simulations.