Observation of non-scalar and logarithmic correlations in 2D and 3D percolation

TL;DR

This study investigates the complex correlation structures in 2D and 3D percolation, confirming theoretical predictions in 2D and exploring new critical exponents and logarithmic behaviors in 3D through extensive simulations.

Contribution

The paper provides high-precision measurements of N-cluster exponents and non-scalar features in 2D and 3D percolation, including the first report of such exponents in 3D.

Findings

Excellent agreement with exact 2D exponents

First report of N-cluster exponents in 3D

Validation of logarithmic structure predictions in 3D

Abstract

Percolation, a paradigmatic geometric system in various branches of physical sciences, is known to possess logarithmic factors in its correlators. Starting from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model with $S_Q$ symmetry, in terms of geometrical clusters, its operator content as $N$-cluster observables has been classified. We extensively simulate critical bond percolation in two and three dimensions and determine with high precision the $N$-cluster exponents and non-scalar features up to $N \! =\! 4$ (2D) and $N \! =\! 3$ (3D). The results are in excellent agreement with the predicted exact values in 2D, while such families of critical exponents have not been reported in 3D, to our knowledge. Finally, we demonstrate the validity of predictions about the logarithmic structure between the energy and two-cluster operators in 3D.