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arXiv:1809.06650 · Physical review. E

Observation of nonscalar and logarithmic correlations in two- and three-dimensional percolation

Xiao-Jun Tan, Romain Couvreur, Youjin Deng, Jesper Lykke Jacobsen

Published May 21, 2019• 11 citations
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Abstract

In bulk percolation, we exhibit operators that insert N clusters with any given symmetry under the symmetric group S_{N}. At the critical threshold, this leads to predictions that certain combinations of two-point correlation functions depend logarithmically on distance, without the usual power law. The behavior under rotations of certain amplitudes of correlators is also determined exactly. All these results hold in any dimension, 2≤d≤6. Moreover, in d=2 the critical exponents and universal logarithmic prefactors are obtained exactly. We test these predictions against extensive simulations of critical bond percolation in d=2 and 3, for all correlators up to N=4 (d=2) and N=3 (d=3), finding excellent agreement. In d=3 we further obtain precise numerical estimates for critical exponents and logarithmic prefactors.

DOI
10.1103/physreve.99.050103

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In bulk percolation, we exhibit operators that insert N clusters with any given symmetry under the symmetric group S_{N}. At the critical threshold, this leads to predictions that certain combinations of two-point correlation functions depend logarithmically on distance, without the usual power law. The behavior under rotations of certain amplitudes of correlators is also determined exactly. All these results hold in any dimension, 2≤d≤6. Moreover, in d=2 the critical exponents and universal logarithmic prefactors are obtained exactly. We test these predictions against extensive simulations of critical bond percolation in d=2 and 3, for all correlators up to N=4 (d=2) and N=3 (d=3), finding excellent agreement. In d=3 we further obtain precise numerical estimates for critical exponents and logarithmic prefactors.
We study N-cluster correlation functions in four- and five-dimensional (4D and 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E 72, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry Ld−1 × ∞, with the linear size up to L = 512 for 4D and 128 for 5D. We determine with a high precision all possible N-cluster exponents, for N = 2 and 3, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with N=2, we obtain the correlation-length critical exponent as 1/ν=1.4610(12) for 4D and 1/ν=1.737(2) for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.
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 carrying out extensive Monte Carlo simulations.
We examine the average cluster distribution as a function of lattice probability for a very small (L=6)(L=6)(L=6) lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of L=6L=6L=6 and L=63L=63L=63 lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.
We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.
We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.
A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.