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Register a new userDirected rigidity and bootstrap percolation in (1+1) dimensions
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Added by
Cristian F. Moukarzel
Publication date
1999
fields
Physics
and research's language is
English
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We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a non-trivial p_c. On the augmented triangular lattice we find, by extensive numerical simulation, that the directed rigidity percolation transition belongs to the same universality class as directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e. the spreading of rigidity from finite clusters close to a non-rigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.
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The percolation behaviour during the deposit formation, when the spanning cluster was formed in the substrate plane, was studied. Two competitive or mixed models of surface layer formation were considered in (1+1)-dimensional geometry. These models are based on the combination of ballistic deposition (BD) and random deposition (RD) models or BD and Family deposition (FD) models. Numerically we find, that for pure RD, FD or BD models the mean height of the percolation deposit $bar h$ grows with the substrate length $L$ according to the generalized logarithmic law $bar hpropto (ln (L))^gamma$, where $gamma=1.0$ (RD), $gamma=0.88pm 0.020$ (FD) and $gamma=1.52pm 0.020$ (BD). For BD model, the scaling law between deposit density $p$ and its mean height $bar h$ at the point of percolation of type $p-p_infty propto bar h^{-1/ u_h}$ are observed, where $ u_h =1.74pm0.02$ is a scaling coefficient. For competitive models the crossover, %in $h$ versus $L$ corresponding to the RD or FD -like behaviour at small $L$ and the BD-like behaviour at large $L$ are observed.
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