# Continuum Percolation Thresholds in Two Dimensions

@article{Mertens2012ContinuumPT, title={Continuum Percolation Thresholds in Two Dimensions}, author={Stephan Mertens and Cristopher Moore}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2012}, volume={86 6 Pt 1}, pages={ 061109 } }

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding…

## 136 Citations

### Two-dimensional continuum percolation models with disks under the generalized Achlioptas process

- PhysicsJournal of Statistical Mechanics: Theory and Experiment
- 2022

We investigate the disk percolation phase transitions in continuum models in two dimensions under four kinds of generalized product rule and four kinds of generalized sum rule, respectively. We study…

### From discrete to continuous percolation in dimensions 3 to 7

- Mathematics, Computer Science
- 2016

The convergence of a discrete model to its continuous limit is controlled by a power-law dependency with a universal exponent θ=3/2, which allows us to estimate the continuous percolation thresholds in a model of aligned hypercubes in dimensions d=3,…,7 with accuracy far better than that attained using any other method before.

### Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

- Mathematics
- 2017

We examine the interplay between anisotropy and percolation, i.e. the spontaneous formation of a system spanning cluster in an anisotropic model. We simulate an extension of a benchmark model of…

### Percolation of overlapping squares or cubes on a lattice

- Computer Science
- 2014

This work proposes a generalization of the excluded volume approximation to discrete systems and uses it to investigate the transition between continuous and discrete percolation, finding a remarkable agreement between the theory and numerical results.

### Self-dual quasiperiodic percolation

- Physics
- 2022

How does the percolation transition behave in the absence of quenched randomness? To address this question, we study a nonrandom self-dual quasiperiodic model of square-lattice bond percolation.…

### Area covered by disks in small-bounded continuum percolating systems: An application to the string percolation model.

- Physics, MathematicsPhysical review. E
- 2019

Modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries are reported, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision.

### Characterizing spatial point processes by percolation transitions

- PhysicsJournal of Statistical Mechanics: Theory and Experiment
- 2022

A set of discrete individual points located in an embedding continuum space can be seen as percolating or non-percolating, depending on the radius of the discs/spheres associated with each of them.…

### Conformal invariance in three dimensional percolation

- Mathematics
- 2015

The aim of the paper is to present numerical results supporting the presence of conformal invariance in three dimensional statistical mechanics models at criticality and to elucidate the geometric…

### A percolation algorithm based on cellular automata

- Computer Science2015 IEEE International Conference on Electro/Information Technology (EIT)
- 2015

An algorithm which determines if there is a spanning clump in lattice or not, which is the largest cluster in the field which that spans the entire network vertically, horizontally or both is presented.

### New algorithm to test percolation conditions within the Newman-Ziff algorithm

- Computer Science
- 2014

An algorithm to verify the existence of both clusters touching boundaries at an arbitrary point and single-loop clusters continuously connecting the opposite boundaries in a percolating system with periodic boundary conditions is proposed within the Newman–Ziff algorithm.

## References

SHOWING 1-10 OF 32 REFERENCES

### Percolation in the canonical ensemble

- Physics
- 2012

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at…

### Scaling and universality in the spanning probability for percolation.

- PhysicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1996

This paper discusses the spanning percolation probability function for three different spanning rules, in general dimensions, with both free and periodic boundary conditions, and finds strong relations among different derivatives of the spanning function with respect to the scaling variables, thus yielding several universal amplitude ratios.

### Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses.

- Computer ScienceThe Journal of chemical physics
- 2012

The present investigation provides additional analytical results for certain cluster statistics, such as the concentration of k-mers and related quantities, and obtains an upper bound on the percolation threshold η(c), and provides accurate analytical estimates of the pair connectedness function and blocking function for any d as a function of density.

### Finite-size scaling in stick percolation.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

A high-efficiency algorithm for Monte Carlo simulations is developed to investigate, with extensive realizations, the finite-size scaling behavior of stick percolation in large-size systems and results indicate that the spanning probability of stickPercolation on square systems with free boundary conditions falls on the same universal scaling function as that for lattice percolations.

### Asymmetry in the percolation thresholds of fully penetrable disks with two different radii.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2007

Simulation of continuum gradient percolation of systems of fully penetrable disks of two different radii improves the measurement of thepercolation threshold for disks of equal radius, and improves the difference from symmetry.

### Continuum percolation threshold for interpenetrating squares and cubes.

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

Simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions for objects whose edges are aligned parallel to one another and randomly oriented objects.

### 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.

### Introduction To Percolation Theory

- Physics
- 1985

Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in…

### Applications of percolation theory

- Materials Science
- 1994

Connectivity as the Essential Physics of Disordered Systems Elements of Percolation Theory Characterization of Porous Media Earthquakes, and Fracture and Fault in Patterns in Heterogeneous Rock…

### The Nature of Computation

- Computer Science, Art
- 2011

The authors explain why the P vs. NP problem is so fundamental, and why it is so hard to resolve, and lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing.