Infinite percolation cluster

According to this model the tunnel current arises due to formation of the infinite percolation cluster of contacting external spheres with i d. The... 

As stated above, the most characteristic feature of percolation is bonding. The dimension of bonding domains (a bonding cluster)—that is, the regions in which it is possible via black bonds to go from one point of the region to another— rapidly increases with the growth of black bond concentration p. When p = pc, an infinite (percolation) cluster spreading over the entire lattice first appears. 

In general, the small-ampUtude oscillatory shear experiment seems to be the more sensitive method for a safe detection of the gel point, formed by the incipient infinite percolated cluster for thermoreversible systems. Sometimes it is difficult to decide which of the TCFs during a temperature scan has the most linear behaviour in the observed delay time window. As it was shown in (Richter et al. 2004b Richter et al. 2005), the effect in rheology is more pronounced, such that 1-2 K... 

Contrary to expectations, on the millisecond time scale the initial drop //< in the luminescence studies of Mays and Ilgenfritz [24] did not increase significantly with the temperature-induced cluster growth but remained constant even when an infinite percolation cluster was present. Furthermore, the observed decays were always exponential (Fig. 9). Evidently, the initial drop no longer reflects the cluster size. The process responsible for it is over within 50 //s and should perhaps rather be looked upon similarly to the transient active sphere part of normal diffusion-controlled decay. The diffusion in this case is a random walk performed by the quencher on a (percolation) cluster. A stretched exponential decay would be expected for a random walk deactivation on a static cluster, as was observed close to the percolation threshold in earlier studies [23,24]. Those measurements were performed over a time window of about 50 //s, which is close to the reported value of the cluster lifetime from electrical birefringence measurements [60]. It is very likely that... 

If the end-to-end distance of a SAW is greater than the correlation length R we find the behavior of a SAW on a normal lattice. On the other hand, if R [Pg.117]

Comparing this with (45) we find y = upc/i p-In Ref. [30] it was noted, that one expects to have two different criticcJ exponents, depending on whether one averages the SAW configurations only on the infinite percolation cluster at Pc or on all clusters. If the SAW is averaged only on the infinite percolation cluster (infinite cluster, IC), we expect to have ... 

To determine the scaling behavior of Hb, one resorts to the probability Poo that an arbitrary site belongs to the infinite percolation cluster. For concentrations p above the percolation threshold Pc, the latter reads... 

We calculated the probability for the largest observed cluster to percolate depending on the hydrogen bond concentration, < hb), under various thermodynamic conditions (Churakov and Kalinichev 2000). From these data, the size of a percolating cluster at r = 1.12 is estimated to be 65 molecules. This value divides cluster size distributions into two parts. One of them represents infinite percolating clusters, while another part represents smaller isolated clusters (Fig. 15). 

Sandvik performed quantum Monte Carlo simulations of the Heisenberg Hamiltonian on the critical infinite percolation cluster (p = pp) using the stochastic series expansion method with operator loop update. 

FIGURE 3.43 A finite size sample with occupied sites (dots) and bounds between them on a square lattice. The samples is shown at different site occupation probability,/ , that is (a) below the percolation threshold, p = 0.2, (b) at the percolation threshold, pc = 0.59, and (c) above the percolation threshold,/ = 0.8. Open symbols in (b) and (c) represent sites that belong to the infinite percolation cluster. 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

We have studied the system (9.1.39) to (9.1.41) by means of the Monte Carlo method on a disordered surface where the active sites form a percolation cluster built at the percolation threshold and also above this threshold [25]. Finite clusters of active sites were removed from the surface to study only the effect of the ramification of the infinite cluster. The phase transition points show strong dependence on the fraction of active sites and on the... 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

Percolation theory is a statistical theory that studies disordered or chaotic systems where the components are randomly distributed in a lattice. A cluster is defined as a group of neighboring occupied sites in the lattice, being considered an infinite or percolating cluster when it extends from one side to the rest of the sides of the lattice, that is, percolates the whole system [38],... 

The percolation threshold pc is defined as the site-filling probability that marks the appearance of the lattice-spanning percolation cluster and the establishment of long-range connectivity. One can introduce the function P(p), called the percolation probability, which has the following significance When the fraction of filled sites is p, P p) is the chance of a randomly chosen site being both filled and part of the infinite cluster, or, in other words, P(,p) is the fraction of the entire system that is taken up by the infinite cluster. 

In the above node-link-blob picture, the percolation cluster is self-similar up to a length scale in the sense that starting from the length scale the links contain blobs (and the dangling ends) which, in turn, are composed of links and blobs (and the dangling ends) up to the lowest scale (of the lattice). This self-similarity extends up to infinite scale at the percolation threshold (where becomes infinitely large). 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

Figure 14. The percolation clusters (a) Isolated cluster, (b) Infinite cluster (schematic).

To quantitatively characterize the percolation cluster as a whole, the notion of infinite cluster density is introduced, Pooip) It is the ratio of the number of the bonds belonging to the infinite cluster, nk to all black bonds on the percolation lattice N ... 

Here, the percolation cluster (infinite cluster) density PKl (p) can be represented as... 

For example, blue bonds are a set of bonds in which current will flow if the percolation cluster is placed between electrodes subject to a potential difference (Fig. 14). The set of blue bonds generates an infinite cluster. 

The set of red bonds consists of those bonds whereby removing one bond disturbs the bonding of the percolation cluster (removing a red bond leads to an open circuit, Fig. 14), while removing a single blue bond does not lead to disturbance of infinite cluster bonding. 

It follows from (6.57, 58) that pmix — /iL can have one or two minima at 0 < x < 1, thus one or two (stable or stable + metastable) states of supercooled liquids can exist. At high temperatures, T > T, and at low temperatures, T < fm, only one equilibrium state exists. If two equilibrium states coexist, they differ by the degree of clusterization. If a clusterized fraction is large enough, the state must be treated as solid one. Indeed, in the system at x > xc 0.16 an infinite (percolated) solid cluster is formed and at (1 — x) > (1 — x)c 0.16 a percolated liquid cluster appears. So, at x > xg 0.84 the mixed state is really a solid with heterophase liquid fluctuations. The temperature at which the stable state with x > xg exists, is the thermodynamic glassing temperature, Tgh. 

More recently, percolation theory and computer simulation of the dissolution process was applied. This latter approach resulted in 2D and 3D percolation thresholds (that is, composition thresholds at which infinite connected paths of the fast dissolving component were formed) as well as in images of the atomic scale disorder induced by dealloying. 3D site percolation thresholds 20 at.% in a fee lattice), leading to an infinite connected cluster of nearest neighbors of less noble atoms, were considered to correlate with the absolute parting limits of alloys with high such as... 

In the case of random fractals, the paradigm and most commonly studied model is percolation (see also Chapter 1 by Chakrabarti). It is important to note that non-trivial modifications of the scaling behavior of SAWs on percolation are believed to occur only at the percolation threshold [5]. Indeed, it is only at criticality, that the infinite critical (the so-called incipient) percolation cluster spans self-simil u structures on all length scales. 

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