$N$-cluster correlations in four- and five-dimensional percolation

TL;DR

This study uses advanced Monte Carlo simulations to precisely determine N-cluster correlation exponents in 4D and 5D percolation, confirming theoretical predictions and revealing new universal amplitude estimates.

Contribution

We reformulate the transfer Monte Carlo algorithm with disjoint-set data structures and accurately compute all N-cluster exponents for N=2,3 in high-dimensional percolation.

Findings

Precisely determined correlation-length exponents: 1/ν=1.4610(12) in 4D and 1.737(2) in 5D.

First estimates of other N-cluster exponents and universal logarithmic amplitude.

Confirmed the validity of logarithmic conformal field theory in high-dimensional percolation.

Abstract

We study $N$-cluster correlation functions in four- and five-dimensional (4D, 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E {\bf 72}, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry $L^{d-1}\times \infty$, 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 $1/\nu \! =\! 1.4610(12)$ for 4D and $1/\nu \! =\! 1.737 (2)$ 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.