# Efficient Monte Carlo algorithm and high-precision results for percolation.

@article{Newman2000EfficientMC, title={Efficient Monte Carlo algorithm and high-precision results for percolation.}, author={Mark E. J. Newman and Robert M. Ziff}, journal={Physical review letters}, year={2000}, volume={85 19}, pages={ 4104-7 } }

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site…

## 323 Citations

### Fast Monte Carlo algorithm for site or bond percolation.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2001

An efficient algorithm is described that can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system.

### Convergence of threshold estimates for two-dimensional percolation.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2002

This work shows that the convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent as predicted previously, and measures the value of this exponent to be 0.90+/-0.02.

### Percolation on two- and three-dimensional lattices.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

A highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff is applied to treat percolation problems to confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.

### Bond percolation on simple cubic lattices with extended neighborhoods.

- PhysicsPhysical review. E
- 2020

The results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111.

### Percolation of randomly distributed growing clusters: finite-size scaling and critical exponents for the square lattice.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

This continuous transition that separates a phase of finite clusters from a phase characterized by the presence of a giant component is studied in detail and is found to belong to a different universality class from the standard percolation transition.

### Percolation of the site random-cluster model by Monte Carlo method.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2015

A site random-cluster model is proposed by introducing an additional cluster weight in the partition function of the traditional site percolation by combining the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon.

### Explosive site percolation and finite-size hysteresis.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2011

Evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolations on lattices, providing that the transitions are continuous and obey the conventional finite size scaling forms is found.

### Percolation of polyatomic species on a simple cubic lattice

- Physics
- 2013

In the present paper, the site-percolation problem corresponding to linear k-mers (containing k identical units, each one occupying a lattice site) on a simple cubic lattice has been studied. The…

### Numerical results for crossing, spanning and wrapping in two-dimensional percolation

- Physics
- 2003

Using a recently developed method to simulate percolation on large clusters of distributed machines [1], we have numerically calculated crossing, spanning and wrapping probabilities in…

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