Surface effects for percolation

Clerc, Jean-Pierre; Carton, Jessy; Roussenq, J.; Stauffer, Dietrich

doi:10.1051/jphyslet:019810042012024900

Cited by 6 publications

(9 citation statements)

References 5 publications

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“…We present some results in section 7, including data on the case with the bulk density fixed at density p near p c (d), where we obtain estimates of the curve σ * , consistent with reference [6].…”

Section: Inhomogeneous Percolationmentioning

confidence: 85%

“…(6) In the supercritical phase the cluster size distribution of the cluster at the origin has sub-exponential decay [2]:…”

Section: The Collection Of Closed Edges Incident With C(x) Is the Permentioning

confidence: 99%

“…Solving for σ gives an estimate of σ * (p) for small p : (20) (the approximation of [6] is different only because they consider a half-space with L 0 being the boundary plane). This result should be good for small values of p in particular, because the assumption that bridges are independent is better at low densities of edges in the bulk lattice.…”

Section: The Phase Boundariesmentioning

confidence: 99%

“…The phase boundary σ * (p) may be estimated for small p in a mean field approximation using the approach in reference [6]. Consider percolation in the defect lattice L 0 , which has critical density σ c = p c (s).…”

Section: The Phase Boundariesmentioning

confidence: 99%

“…Surface phenomena in percolation have also received considerable attention [6,8,9,10,11,12]. This is a model of gelation along a defect plane or surface, and was also interpreted as a model of branch polymer adsorption in bulk [11] -results in references [8,11] suggest that a surface transition is absent in two dimensional models.…”

Section: Introductionmentioning

confidence: 99%

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Phase Diagram of Inhomogeneous Percolation with a Defect Plane

2014

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Let L be the d-dimensional hypercubic lattice and let L 0 be an s-dimensional sublattice, with 2 ≤ s < d. We consider a model of inhomogeneous bond percolation on L at densities p and σ, in which edges in L\L 0 are open with probability p, and edges in L 0 open with probability σ. We generalizee several classical results of (homogeneous) bond percolation to this inhomogeneous model. The phase diagram of the model is presented, and it is shown that there is a subcritical regime for σ < σ * (p) and p < p c (d) (where p c (d) is the critical probability for homogeneous percolation in L), a bulk supercritical regime for p > p c (d), and a surface supercritical regime for p < p c (d) and σ > σ * (p). We show that σ * (p) is a strictly decreasing function for p ∈ [0, p c (d)], with a jump discontinuity at p c (d). We extend the Aizenman-Barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase. We prove that the cluster size distribution decays exponentially in the subcritical phase, and sub-exponentially in the supercritical phases. For a model of lattice animals with a defect plane, the free energy is related to functions of the inhomogeneous percolation model, and we show how the percolation transition implies a non-analyticity in the free energy of the animal model. Finally, we present simulation estimates of the critical curve σ * (p).

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Section: Inhomogeneous Percolationmentioning

confidence: 85%

“…(6) In the supercritical phase the cluster size distribution of the cluster at the origin has sub-exponential decay [2]:…”

Section: The Collection Of Closed Edges Incident With C(x) Is the Permentioning

confidence: 99%

Section: The Phase Boundariesmentioning

confidence: 99%

Section: The Phase Boundariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%