# From discrete to continuous percolation in dimensions 3 to 7

@article{Koza2016FromDT, title={From discrete to continuous percolation in dimensions 3 to 7}, author={Zbigniew Koza and Jakub Pola}, journal={Journal of Statistical Mechanics: Theory and Experiment}, year={2016}, volume={2016} }

We propose a method of studying the continuous percolation of aligned objects as a limit of a corresponding discrete model. We show that the convergence of a discrete model to its continuous limit is controlled by a power-law dependency with a universal exponent θ=3/2. This 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. We also report improved values of…

## 25 Citations

### Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2022

We study the onset of percolation of overlapping discrete hyperspheres on hypercubic lattices in dimension D = 3, 4, 5. Taking the continuum limit of the thresholds for discrete hyperspheres we…

### Critical p=1/2 in percolation on semi-infinite strips.

- MathematicsPhysical review. E
- 2019

It is argued that the probability that a cluster touches the three sides of a planar system at the percolation threshold has a continuous limit of 1/2 and that this limit is universal for planar systems.

### Dynamics around the site percolation threshold on high-dimensional hypercubic lattices.

- PhysicsPhysical review. E
- 2019

The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d{u}.

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

- PhysicsFrontiers of Physics
- 2020

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…

### 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.

### Jamming and percolation of k2-mers on simple cubic lattices

- PhysicsPhysical review. E
- 2019

A complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the 3D random percolations, regardless of the size k considered.

### Random site percolation on honeycomb lattices with complex neighborhoods.

- PhysicsChaos
- 2022

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical…

### Percolation of sites not removed by a random walker in d dimensions.

- MathematicsPhysical review. E
- 2019

This work systematically explore dependence of the probability Π_{d}(L,u) of percolation (existence of a spanning cluster) of sites not removed by the RW on L and u, which shows the concentration of unvisited sites decays exponentially with increasing u, while the visited sites are highly correlated.

### Percolation in a triangle on a square lattice

- Mathematics, Physics
- 2021

Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the…

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