Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...
New high-density series data for the mean number, percolation probability and 'susceptibility' of finite clusters are presented for bond and site percolation on four standard three-dimensional lattices. A Pade approximant analysis of both high- and low-density series makes particular use of the rather precise unbiased estimates of the percolation threshold, p c , obtained recently by Heermann and Stauffer (1981) for the simple cubic lattice. For this lattice the authors obtain the biased estimate gamma =1.73+or-0.03 for the site problem and a similar estimate but with larger uncertainties for the bond problem. Such a value is significantly larger than earlier series estimates. Assuming gamma to be universal they obtain precise, although biased, estimates of p c for both bond and site percolation on all four lattices. Using the bond estimates of p c they find an overall biased estimate of beta =0.454+or-0.008 for bond percolation on all three-dimensional lattices. (The corresponding site problem requires further study.) Scaling estimates of other critical exponents are alpha =-0.64+or-0.05, delta =4.81+or-0.14, Delta =2.18+or-0.04, nu =0.88+or-0.02 and nu =0.03+or-0.03.
The superfluid to insulator quantum phase transition of a three-dimensional particle-hole symmetric system of disordered bosons is studied. To this end, a site-diluted quantum rotor Hamiltonian is mapped onto a classical (3+1)-dimensional
XY model with columnar disorder and analyzed by means of large-scale Monte Carlo simulations. The superfluid-Mott insulator transition of the clean, undiluted system is in the four-dimensional
XY universality class and shows mean-field critical behavior with logarithmic corrections. The clean correlation length exponent $\ensuremath{\nu}=1/2$ violates the Harris criterion, indicating that disorder must be a relevant perturbation. For nonzero dilutions below the lattice percolation threshold of
pc=0.688392, our simulations yield conventional power-law critical behavior with dilution-independent critical exponents
z=1.67(6), $\ensuremath{\nu}=0.90(5)$, $\ensuremath{\beta}/\ensuremath{\nu}=1.09(3)$, and $\ensuremath{\gamma}/\ensuremath{\nu}=2.50(3)$. The critical behavior of the transition across the lattice percolation threshold is controlled by the classical percolation exponents. Our results are discussed in the context of a classification of disordered quantum phase transitions, as well as experiments in superfluids, superconductors, and magnetic systems.