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Register a new userThe size of a pond in 2D invasion percolation
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We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n from the probability that in critical Bernoulli percolation the radius of an open cluster is larger than n. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond.
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In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.
We study the boundary effects in invasion percolation with and without trapping. We find that the presence of boundaries introduces a new set of surface critical exponents, as in the case of standard percolation. Numerical simulations show a fractal dimension, for the region of the percolating cluster near the boundary, remarkably different from the bulk one. We find a logarithmic cross-over from surface to bulk fractal properties, as one would expect from the finite-size theory of critical systems. The distribution of the quenched variables on the growing interface near the boundary self-organises into an asymptotic shape characterized by a discontinuity at a value $x_c=0.5$, which coincides with the bulk critical threshold. The exponent $tau^{sur}$ of the boundary avalanche distribution for IP without trapping is $tau^{sur}=1.56pm0.05$; this value is very near to the bulk one. Then we conclude that only the geometrical properties (fractal dimension) of the model are affected by the presence of a boundary, while other statistical and dynamical properties are unchanged. Furthermore, we are able to present a theoretical computation of the relevant critical exponents near the boundary. This analysis combines two recently introduced theoretical tools, the Fixed Scale Transformation (FST) and the Run Time Statistics (RTS), which are particularly suited for the study of irreversible self-organised growth models with quenched disorder. Our theoretical results are in rather good agreement with numerical data.
In this paper we study the properties of the Barabasi model of queueing under the hypothesis that the number of tasks is steadily growing in time. We map this model exactly onto an Invasion Percolation dynamics on a Cayley tree. This allows to recover the correct waiting time distribution $P_W(tau)sim tau^{-3/2}$ at the stationary state (as observed in different realistic data) and also to characterize it as a sequence of causally and geometrically connected bursts of activity. We also find that the approach to stationarity is very slow.
The main purpose of this work is to simulate two-phase flow in the form of immiscible displacement through anisotropic, three-dimensional (3D) discrete fracture networks (DFN). The considered DFNs are artificially generated, based on a general distribution function or are conditioned on measured data from deep geological investigations. We introduce several modifications to the invasion percolation (MIP) to incorporate fracture inclinations, intersection lines, as well as the hydraulic path length inside the fractures. Additionally a trapping algorithm is implemented that forbids any advance of the invading fluid into a region, where the defending fluid is completely encircled by the invader and has no escape route. We study invasion, saturation, and flow through artificial fracture networks, with varying anisotropy and size and finally compare our findings to well studied, conditioned fracture networks.
In J. Shao et al., PRL 103, 108701 (2009) the authors claim that a model with majority rule coarsening exhibits in d=2 a percolation transition in the universality class of invasion percolation with trapping. In the present comment we give compelling evidence, including high statistics simulations on much larger lattices, that this is not correct. and that the model is trivially in the ordinary percolation universality class.
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