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

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 $Q\to 1$ 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.

arXiv comment10 pages, 7 figures

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• Topological effects and conformal invariance in long-range correlated random surfaces
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories
• Universal correlations in percolation
• Universality of closed nested paths in two-dimensional percolation
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice