## Inhomogeneous percolation problems and incipient infinite clusters

The authors consider inhomogeneous percolation models with density p c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod - lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model, they prove that in d=2, the borderline value of lambda is lambda b=1/v. If lambda >1/v then, with probability one, there is no infinite cluster, while if lambda <1/v then, with positive probability, the origin is part of an infinite cluster. This result sheds some light on numerical and theoretical predictions of certain properties of incipient infinite clusters. Furthermore, for d>2, the models studied suggest what sort of 'incipient objects' should be examined in random surface models.

### Duke Scholars

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- Mathematical Physics
- 02 Physical Sciences
- 01 Mathematical Sciences

### Citation

*Journal of Physics A: Mathematical and General*,

*20*(6), 1521–1530. https://doi.org/10.1088/0305-4470/20/6/034

*Journal of Physics A: Mathematical and General*20, no. 6 (December 1, 1987): 1521–30. https://doi.org/10.1088/0305-4470/20/6/034.

*Journal of Physics A: Mathematical and General*, vol. 20, no. 6, Dec. 1987, pp. 1521–30.

*Scopus*, doi:10.1088/0305-4470/20/6/034.

## Published In

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## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Mathematical Physics
- 02 Physical Sciences
- 01 Mathematical Sciences