The geometry of percolation fronts in two-dimensional lattices with spatially varying densities
Percolation theory is usually applied to lattices with a uniform probability p that a site is occupied or that a bond is closed. The more general case, where p is a function of the position x , has received less attention. Previous studies with long-range spatial variations in p ( x ) have only inve...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2012-01-01
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| Series: | New Journal of Physics |
| Online Access: | https://doi.org/10.1088/1367-2630/14/10/103019 |
| _version_ | 1797751635951222784 |
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| author | Michael T Gastner Beáta Oborny |
| author_facet | Michael T Gastner Beáta Oborny |
| author_sort | Michael T Gastner |
| collection | DOAJ |
| description | Percolation theory is usually applied to lattices with a uniform probability p that a site is occupied or that a bond is closed. The more general case, where p is a function of the position x , has received less attention. Previous studies with long-range spatial variations in p ( x ) have only investigated cases where p has a finite, non-zero gradient at the critical point p _c . Here we extend the theory to two-dimensional cases in which the gradient can change from zero to infinity. We present scaling laws for the width and length of the hull (i.e. the boundary of the spanning cluster). We show that the scaling exponents for the width and the length depend on the shape of p ( x ), but they always have a constant ratio 4/3 so that the hull's fractal dimension D = 7/4 is invariant. On this basis, we derive and verify numerically an asymptotic expression for the probability h ( x ) that a site at a given distance x from p _c is on the hull. |
| first_indexed | 2024-03-12T16:51:25Z |
| format | Article |
| id | doaj.art-3801f51375b9454f8cf20b86daccf2c0 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:51:25Z |
| publishDate | 2012-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-3801f51375b9454f8cf20b86daccf2c02023-08-08T11:07:49ZengIOP PublishingNew Journal of Physics1367-26302012-01-01141010301910.1088/1367-2630/14/10/103019The geometry of percolation fronts in two-dimensional lattices with spatially varying densitiesMichael T Gastner0Beáta Oborny1Department of Mathematics, Complexity and Networks Programme, Imperial College London, South Kensington Campus, London SW7 2AZ, UK; Department of Engineering Mathematics, University of Bristol , Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UKDepartment of Plant Taxonomy, Ecology and Theoretical Biology, Eötvös Lorand University , Pázmány sétány 1/C, Budapest H-1117, HungaryPercolation theory is usually applied to lattices with a uniform probability p that a site is occupied or that a bond is closed. The more general case, where p is a function of the position x , has received less attention. Previous studies with long-range spatial variations in p ( x ) have only investigated cases where p has a finite, non-zero gradient at the critical point p _c . Here we extend the theory to two-dimensional cases in which the gradient can change from zero to infinity. We present scaling laws for the width and length of the hull (i.e. the boundary of the spanning cluster). We show that the scaling exponents for the width and the length depend on the shape of p ( x ), but they always have a constant ratio 4/3 so that the hull's fractal dimension D = 7/4 is invariant. On this basis, we derive and verify numerically an asymptotic expression for the probability h ( x ) that a site at a given distance x from p _c is on the hull.https://doi.org/10.1088/1367-2630/14/10/103019 |
| spellingShingle | Michael T Gastner Beáta Oborny The geometry of percolation fronts in two-dimensional lattices with spatially varying densities New Journal of Physics |
| title | The geometry of percolation fronts in two-dimensional lattices with spatially varying densities |
| title_full | The geometry of percolation fronts in two-dimensional lattices with spatially varying densities |
| title_fullStr | The geometry of percolation fronts in two-dimensional lattices with spatially varying densities |
| title_full_unstemmed | The geometry of percolation fronts in two-dimensional lattices with spatially varying densities |
| title_short | The geometry of percolation fronts in two-dimensional lattices with spatially varying densities |
| title_sort | geometry of percolation fronts in two dimensional lattices with spatially varying densities |
| url | https://doi.org/10.1088/1367-2630/14/10/103019 |
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