Invasion Percolation Clusters Research Articles

Scaling of the cumulative weights of the invasion percolation cluster on a branching process tree

Scaling of the cumulative weights of the invasion percolation cluster on a branching process tree

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [9]. We describe our results in terms of a sequence of parameters $(s_n)_{n\geq 1}$ that quantifies the extreme-value behavior of small weights, and that describes different universality classes for first passage percolation on the complete graph. We consider both $n$-independent as well as $n$-dependent edge weights. The simplest example consists of edge weights of the form $E^{s_n}$, where $E$ is an exponential random variable with mean 1. In this paper, we investigate the case where $s_n\rightarrow \infty$, and focus on the local neighborhood of a vertex. We establish that the smallest-weight tree of a vertex locally converges to the invasion percolation cluster on the Poisson weighted infinite tree. In addition, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices.

Subdiffusivity of random walk on the 2D invasion percolation cluster

We derive quenched subdiffusive lower bounds for the exit time τ(n) from a box of size n for the simple random walk on the planar invasion percolation cluster. The first part of the paper is devoted to proving an almost sure analogue of H. Kesten’s subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on the intrinsic distance in these graphs and general inequalities for reversible Markov chains. In the second part of the paper, we present a sharpening of Kesten’s original argument, leading to an explicit almost sure lower bound for τ(n) in terms of percolation arm exponents. The methods give τ(n)≥n2+ϵ0+κ, where ϵ0>0 depends on the intrinsic distance and κ can be taken to be 5384 on the hexagonal lattice.

Scaling limit of the invasion percolation cluster on a regular tree

We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive stochastic processes. This convergence allows us to recover and make precise certain asymptotic results for the IPC. In particular, we relate the limit of the rescaled level sets of the IPC to the local time of the scaled height function.

Fractal Phase Distribution and Drying: Impact on Two-Phase Zone Scaling and Drying Time Scale Dependence

In an article published in 2008,[ 1 ] Professor A.R. Mujumdar and his colleagues reviewed some applications of fractal concept on drying. As a modest continuation to this article, we give an overview on three drying-related issues where fractal aspects are present. First, we discuss within the framework of the theory of invasion percolation in a gradient the characteristic lengths that determine the extent of the hydraulically connected region during drying. It is pointed out that the scaling of this region is fundamentally different in 2D and in 3D, owing to the different percolation properties in 2D and 3D. In particular, it is shown that the fractal region only represents a small region of a drying front in 3D systems. Then a situation is described where fractal porous structures form as a result of an evaporation process. Finally, we consider drying in systems characterized by an initial fractal distribution of the liquid phase (invasion percolation cluster), a situation expected to happen in PEM fuel cells, and explore the size-dependent property of the overall drying time from pore network simulations.

The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions

This note is motivated by results of Angel, Goodman, den Hollander and Slade (2008) and Da, Sapozhnikov and Vagvolgyi (2009) about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on trees and in two dimensions.

Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IIC's). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the distribution of the number of outlets in an annulus. This result leads to almost sure bounds for the number of outlets in a box B(2^n) and for the decay rate of the weight of the kth outlet to p_c. We then prove existence of multiple-armed IIC measures for any number of arms and for any color sequence which is alternating or monochromatic. We use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin.

Invasion of a sticky random solid: Self-established potential gradient, phase separation, and criticality

Invasion of a sticky random solid by an aqueous solution is modeled through a chemical reaction. In this reaction, the solid elements dissolve in the solution and redeposit back on the rough interface. A self-established potential gradient (SEPG) in the binding energy of the solid is developed spontaneously and the system gets phase separated into "hard" and "soft" solids. The solution profile is found drifted slowly into the solid by the SEPG with a constant velocity. The system tunes itself to the percolation threshold in the steady state. In the steady state, the system is found consisting of finite clusters of solution molecules followed by a path of redeposited solid as an invasion percolation cluster. A diffusive growth of the interface and the solution inside the solid is found to occur. The nonequilibrium steady state of this dynamical system is found critical and characterized by a power-law distribution of cluster size with an exponent approximately -2 .

The strange man in random networks of automata

We have performed computer simulations of Kauffman’s automata on several graphs, such as the regular square lattice and invasion percolation clusters, in order to investigate phase transitions, radial distributions of the mean total damage (dynamical exponent) and propagation speeds of the damage when one adds a damaging agent, nicknamed “strange man”. Despite the increase in the damaging efficiency, we have not observed any appreciable change of the transition threshold to chaos neither for the short-range nor for the small-world case on the square lattices when the strange man is added, in comparison to when small initial damages are inserted in the system. Particularly, we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters (high boundary-saturation clusters). It is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters.

The shape of invasion percolation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for bothnon-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two differentanisotropy quantifiers, the anisotropy parameter and the asphericity, are used for probingthe degree of anisotropy of clusters. We observe that, in spite of the difference in scalingproperties of NTIP and TIP, there is no difference in the values of anisotropyquantifiers of these processes. Furthermore, we find that, in completely randommedia, the invasion percolation clusters are on average slightly less isotropic thanstandard percolation clusters. Introducing isotropic long-range correlations into themedia reduces the isotropy of the invasion percolation clusters. The effect is morepronounced for the case of persisting long-range correlations. The implicationof boundary conditions on the shape of clusters is another subject of interest.Compared to the case of free boundary conditions, IP clusters of conventionalrectangular geometry turn out to be more isotropic. Moreover, we see that inconventional rectangular geometry the NTIP clusters are more isotropic than the TIPclusters.

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its r-point function for any r≥2 and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

Modeling collective behavior of molecules in nanoscale direct deposition processes

We present a theoretical model describing the collective behavior of molecules in nanoscale direct deposition processes such as dip-pen nanolithography. We show that strong intermolecular interactions combined with nonuniform substrate-molecule interactions can produce various shapes of molecular patterns including fractal-like structures. Computer simulations reveal circular and starlike patterns at low and intermediate densities of preferentially attractive surface sites, respectively. At large density of such surface sites, the molecules form a two-dimensional invasion percolation cluster. Previous experimental results showing anisotropic patterns of various chemical and biological molecules correspond to the starlike regime [P. Manandhar et al., Phys. Rev. Lett. 90, 115505 (2003); J.-H. Lim and C. A. Mirkin, Adv. Mater. (Weinheim, Ger.) 14, 1474 (2002); D. L. Wilson et al., Proc. Natl. Acad. Sci. U.S.A. 98, 13660 (2001); M. Su et al., Appl. Phys. Lett. 84, 4200 (2004); R. McKendry et al., Nano Lett. 2, 713 (2002); H. Zhou et al., Appl. Surf. Sci. 236, 18 (2004); G. Agarwal et al., J. Am. Chem. Soc. 125, 580 (2003)].

Nonequilibrium capillary pressure and relative permeability curves of porous media

AbstractHistory matching of unsteady immiscible displacement experiments performed on model porous media reveals that nonequilibrium capillary pressure and relative permeability functions are sensitive to the dominating flow pattern. As the capillary number increases, the network displacement pattern changes gradually from invasion percolation cluster to stable displacement. For primary drainage, the capillary pressure and relative permeability curves are increasing functions of the capillary number. The dependence of the relevant macroscopic parameters on the capillary number is consistent with power laws derived from scaling arguments of gradient percolation theory, which describes the growth of the displacement front. The material coefficient of the thermodynamic theory of capillary pressure changes mildly with fluid saturation and is a decreasing function of the capillary number.

Invasion Percolation and the Incipient Infinite Cluster in 2D

We establish two links between two-dimensional invasion percolation and Kesten's incipient infinite cluster (IIC). We first prove that the k th moment of the number of invaded sites within the box [−n, n]×[−n, n] is of order (n 2π n ) k , for k≥1, where π n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n. This improves a result of Y. Zhang. We show that the size of the invaded region, when scaled by n 2π n , is tight. Secondly, we prove that the invasion cluster looks asymptotically like the IIC, when viewed from an invaded site v, in the limit |v|→∞. We also establish this when an invaded site v is chosen at random from a box of radius n, and n→∞.

Pattern Structure and Properties of Invasion Percolation inPorous Media

The percolation clusters with varying occupy probability areconstructed. Invasion percolation (IP) in percolation cluster isinvestigated by means of IP algorithm without trapping. The patternconstructions of IP in percolation clusters are obviously differentfrom that of viscous fingering (VF) in percolation clusters. Thefractal dimension Df of IP increases with increasing the percolation probability P. The geometry and the topology of theporous media have strong effects on the pattern structure of IP. Forlarge M (the node numbers occupied by the injected fluid), the rescaledvalue is Rg/M1/Df, which asymptoticallyapproaches a constant value, and Rg is the gyrationradius of IP cluster. Moreover, the chemical dimensionDl and the shortest path exponent dmin are obtained.

Nearly space-filling fractal networks of carbon nanopores.

Small-angle x-ray scattering, nitrogen adsorption, and scanning tunneling microscopy show that a series of activated carbons host an extended fractal network of channels with dimension D(p) = 2.8-3.0 (pore fractal), channel width 15-20 A (lower end of scaling), network diameter 3000-3400 A (upper end of scaling), and porosity of 0.3-0.6. We interpret the network as a stack of quasiplanar invasion percolation clusters, formed by oxidative removal of walls between closed voids of diameter of approximately 10 A and held in registry by fibrils of the biological precursor, and point out unique applications.

Aging kinetics of porous media due to freezing-thawing cycles

The two-dimensional Ausloos et al. model of fluid invasion, freezing and thawing in a porous medium is elaborated upon and investigated in order to take into account the pore volume redistribution and conservation during freezing. The results are qualitatively different from previous work, since the damaged pore sizes are found to be much less than the possible maximum value and is reached after a large number of invasion-freezing-thawing cycles, e.g. the material is “slowly damaged”. The pore size distribution is thus found in better agreement with expected practical findings. The successive invasion percolation clusters are still found to be self-avoiding with aging. The cluster size decreases with a power law as a function of invasion-frost-thaw iterations. The aging kinetics is also discussed through the normalized totally invaded pore volume.

Optimal paths and growth processes

Interfaces in systems with strong quenched disorder are fractal and are thus in a different universality class than the self-affine interfaces found in systems with weak quenched disorder. The geometrical properties of strands arising in loopless invasion percolation clusters, in loopless Eden growth clusters, and in the ballistic growth process are studied and their universality classes are identified.

Invasion percolation in fractal fractures

Invasion percolation (IP) with trapping was studied on two-dimensional substrates with a correlated distribution of invasion thresholds. The substrates represented the void spaces of variable-aperture fracture models that were formed by (2+1)-dimensional self-affine rough surfaces and their replicas, displaced relative to each other. Void spaces generated in this manner were self-affine on length scales less than the horizontal displacement r d , and uncorrelated on larger length scales. The geometry of the IP clusters reflected the void space geometry, with different scaling behavior on length scales below and beyond r d .

Fragmentation and migration of invasion percolation clusters: Experiments and simulations

Experimental studies of two fluid displacement processes in porous media involving extensive fragmentation of invasion percolationlike structures are described. In the first process, a two-dimensional porous cell saturated with a wetting fluid was slowly invaded by air. The air formed a fractal structure that fragmented when the pressure of the wetting fluid increased and the air was driven out of the system. In the second process, a fractal air structure migrated through a two-dimensional porous medium saturated with wetting fluid. The structure was driven by increasing buoyancy forces and fragmented. The fragments migrated, fragmented, and coalesced with other fragments. The processes were simulated using new site-bond invasion percolation models that captured the displacement mechanisms and reproduced the fragmentation events, and good agreement was found. In both processes, the fractal dimensionality of the fragments was equal to the dimensionality D\ensuremath{\approx}1.82 of the initial invasion percolationlike structures. The fragment size distributions measured in both processes and the dynamics of the migration process could be described by simple scaling forms.