Infinite Cluster Research Articles

High-Dimensional Incipient Infinite Clusters Revisited

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls.

Time-resolved photoluminescence spectroscopy of localized exciton magnetic polarons in Cd0.70Mn0.30Te spin glass compound

We have investigated dynamics of different localized exciton magnetic polarons (LEMPs) in Cd0.70Mn0.30Te spin glass (SG) compound below the freezing temperature Tf in the crystal regions, where various microscopic magnetic spin states (MMSSs), namely, “loose” spins, finite, and infinite clusters, are formed. It was shown that there is a broad distribution of the LEMPs lifetimes. The presence of the long-lived LEMPs is caused by the admixture of the optically active bright exciton states to the dark exciton states, i.e., the “brightening” of the dark LEMPs which exist along with the bright LEMPs. The lifetimes of the dark LEMPs correspond to hundreds of nanoseconds. It was found that the time decay of photoluminescence band intensity is approximated by the sum of two functions: a single exponential function and the Kohlrausch–Williams–Watts stretched exponential function. The stretched exponential function describes the recombination processes of the LEMPs formed in the crystal regions of the finite clusters as well as the infinite cluster. This reflects the appearance of spatially heterogeneous dynamics in Cd0.70Mn0.30Te SG compound below Tf which is due to the disorder in the spin distribution caused by the formation of different MMSSs.

Percolation effects in the capacitive properties of metal-oxide varistors in the range of high voltage

C–V characteristics of ZnO -based ceramic structures used in manufacturing high-voltage and low-voltage varistors of different chemical compositions and manufacturing techniques have been investigated. A correlation between the intensity of electric field corresponding to transition of the C–V characteristics to the negative capacitances and average sizes of grains of a varistor structure has been established. Obtained data have been interpreted with the use of notions of the percolation theory of electric conductivity. The Shklovskii–De Gennes model has been used. It has been shown that on the highly nonlinear segment of C–V characteristics of a varistor structure, the size of an infinite cluster are limited to several intercrystallite potential barriers. This result is observed in all kinds of investigated varistor ceramics.

On chiral symmetry breaking, topology and confinement

We start with the relation between the chiral symmetry breaking and gauge field topology. New lattice result further enhance the notion of Zero Mode Zone, a very narrow strip of states with quasizero Dirac eigenvalues. Then we move to the issue of "origin of mass" and Brown-RHo scaling: a number of empirical facts contradicts to the idea that masses of quarks and such hadrons as $\rho,N$ decrease near $T_c$. We argue that while at $T=0$ the main contribution to the effective quark mass is chirally odd $m_{\snchi}$, near $T_c$ it rotates to chirally-even component $m_\chi$, because "infinite clusters" of topological solitons gets split into finite ones. Recent progress in understanding of topology require introduction of nonzero holonomy $<A_0>\neq 0$, which splits instantons into $N_c$ (anti)selfdual "instanton-dyons". Qualitative progress, as well as first numerical studios of the dyon ensemble are reported. New connections between chiral symmetry breaking and confinement are recently understood, since instanton-dyons generates holonomy potential with a minimum at confining value, if the ensemble is dense enough.

Random Walk on the High-Dimensional IIC

We study the asymptotic behavior of the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by Kumagai and Misumi (J Theor Probab 21:910–935, 2008). We do this by getting bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander–Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation.

Existence of a Non-Averaging Regime for the Self-Avoiding Walk on a High-Dimensional Infinite Percolation Cluster

Let \(Z_N\) be the number of self-avoiding paths of length \(N\) starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on \({\mathbb Z} ^d\) with parameter \(p>p_c({\mathbb Z} ^d)\). The object of this paper is to study the connective constant of the dilute lattice \(\limsup _{N\rightarrow \infty } Z_N^{1/N}\), which is a non-random quantity. We want to investigate if the inequality \(\limsup _{N\rightarrow \infty } (Z_N)^{1/N} \le \lim _{N\rightarrow \infty } {\mathbb E} [Z_N]^{1/N}\) obtained with the Borel–Cantelli Lemma is strict or not. In other words, we want to know if the quenched and annealed versions of the connective constant are equal. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when \(d\) is sufficiently large there exists \(p^{(2)}_c>p_c\) such that the inequality is strict for \(p\in (p_c,p^{(2)}_c)\).

A Fast Algorithm for Invasion Percolation

We present a computationally fast Invasion Percolation (IP) algorithm. IP is a numerical approach for generating realistic fluid distributions for quasi-static (i.e., slow) immiscible fluid invasion in porous media. The algorithm proposed here uses a binary-tree data structure to identify the site (pore) connected to the invasion cluster that is the next to be invaded. Gravity is included. Trapping is not explicitly treated in the numerical examples but can be added, for example, using a Hoshen–Kopelman algorithm. Computation time to percolation for a 3D system having \(N\) total sites and \(M\) invaded sites at percolation goes as \(O(M \log M)\) for the proposed binary-tree algorithm and as \(O(M N)\) for a standard implementation of IP that searches through all of the uninvaded sites at each step. The relation between \(M\) and \(N\) is \(M = N^{D/E}\), where \(D\) is the fractal dimension of an infinite cluster and \(E\) is Euclidean space dimension. In numerical practice, on finite-sized cubic lattices with invasion structures influenced by the injection boundary and boundary conditions lateral to the flow direction, we observe the scaling \(M = N^{0.852}\) in 3D (valid through the second decimal place) instead of \(M= N^{0.843}\) based on the infinite cluster fractal dimension \(D=2.53\).

Percolation of finite clusters and infinite surfaces

AbstractTwo related issues are explored for bond percolation on ${\mathbb{Z}^d$ (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point psurf satisfies psurf ≥ pfin.

A Systematic Research on the Synthesis, Structures, and Application in Photocatalysis of Cluster-Based Coordination Complexes

To search for effective photocatalysts in the field of cluster-based coordination complexes, we synthesized {[Cd2(dcpp)(beb)2 (H2O)]·H2O}n (1), [Cd2(dcpp)(bmb)(H2O)2]n (2), {[Cd2(dcpp)(2,2′-bipy)(H2O)2]·2H2O}n (3), and {[Cd2(dcpp)(4,4′-bipy)0.5(H2O)3]·H2O}n (4) by utilizing a butterfly-shaped multidentate carboxylic acid (4,5-di(4′-carboxylphenyl)phthalic acid) (H4dcpp)) and N-donor ligands (1,4-bis(2-ethylbenzimidazol-1-ylmethyl) benzene (beb), 1,4-bis(2-methylbenzimidazol-1-ylmethyl) benzene (bmb), 2,2′-bipyridine (2,2′-bipy), 4,4′-bipyridine (4,4′-bipy)). Complexes 1–4 are constructed based on different SBUs. Complexes 1 and 2 possess three-dimensional (3D) pillar-layered frameworks based on noncluster-type SBUs and dinuclear Cd2 clusters units, respectively. Complex 3 is a 3D porous structure building on Cd4 clusters SBUs. Complex 4 is constructed from rare infinite botryoid-like Cd cluster chains and exhibits a 3D complicated framework. The research on optical energy gaps of complexes 1–4 indicates that these complexes are potential semiconductive materials. Moreover, complexes 1–4 are applied to catalyze the reaction of photocatalytic degradation of methylene orange (MO) under high-pressure mercury lamp irradiation. Excitingly, they exhibit good photocatalytic properties in the presence of H2O2, and the degradation rates for 1–4 increase from 43% (without photocatalyst) to 97, 78, 85, and 67%, respectively, after 100 min of irradiation. The photoluminescent properties of complexes 1–4 were also studied.

Bernoulli and self-destructive percolation on non-amenable graphs

In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as $p\searrow p_c$. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.

Uniqueness of the infinite homogeneous cluster in the 1-2 model

A 1-2 model configuration is a subset of edges of the hexagonal lattice such that each vertex is incident to one or two edges. We prove that for any translation-invariant Gibbs measure of 1-2 model, almost surely the infinite homogeneous cluster is unique.

1-2 model, dimers, and clusters

The 1-2 model is a probability measure on subgraphs of the hexagonal lattice, satisfying the condition that the degree of present edges at each vertex is either 1 or 2. We prove that for any translation-invariant Gibbs measure of the 1-2 model on the plane, almost surely there are no infinite paths. Using a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and perfect matchings on a decorated graph, we construct an explicit translation-invariant measure $P$ for 1-2 model configurations on the bi-periodic hexagonal lattice embedded into the whole plane. We prove that the behavior of infinite clusters is different for small and large local weights, which shows the existence of a phase transition.

Further results on consensus formation in the Deffuant model

The so-called Deffuant model describes a pattern for social interaction, in which two neighboring individuals randomly meet and share their opinions on a certain topic, if their discrepancy is not beyond a given threshold $\theta$. The major focus of the analyses, both theoretical and based on simulations, lies on whether these single interactions lead to a global consensus in the long run or not. First, we generalize a result of Lanchier for the Deffuant model on $\mathbb{Z}$, determining the critical value for $\theta$ at which a phase transition of the long term behavior takes place, to other distributions of the initial opinions than i.i.d. uniform on $[0,1]$. Then we shed light on the situations where the underlying line graph $\mathbb{Z}$ is replaced by higher-dimensional lattices $\mathbb{Z}^d,\ d\geq2$, or the infinite cluster of supercritical i.i.d. bond percolation on these lattices.

Cover times for sequences of reversible Markov chains on random graphs

We provide conditions that classify cover times for sequences of random walks on random graphs into two types: One type (Type 1) is the class of cover times that are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (Type 2) is the class of cover times that are of the order of the maximal hitting times. The conditions are described by some parameters determined by the underlying graphs: the volumes, the diameters with respect to the resistance metric, the coverings or packings by balls in the resistance metric. We apply the conditions to and classify a number of examples, such as supercritical Galton-Watson trees, the incipient infinite cluster of a critical Galton-Watson tree and the Sierpinski gasket graph.

Classification of transport backbones of complex networks

Transport properties in random and scale-free (SF) networks are studied by analyzing the betweenness centrality (BC) distribution P(B) in the minimum spanning trees (MSTs) and infinite incipient percolation clusters (IIPCs) of the networks. It is found that P(B) in MSTs scales as P(B)∼B(-δ). The obtained values of δ are classified into two different categories, δ≃1.6 and δ≃2.0. Using the mapping between BC and the branch size of tree structures, it is proved that δ in MSTs which are close to critical trees is 1.6. In contrast, δ in MSTs which are supercritical trees is shown to be 2.0. We also find δ=1.5 in IIPCs, which is a natural result because IIPC is physically critical. Based on the results in MSTs, a physical reason why δ≥2 in the original networks is suggested.

Dispersion transition and the nonergodicity of the disordered nanoporous medium-nonwetting liquid system

The experiments in which a nonwetting liquid does not flow from a disordered nanoporous medium are described. The outflow is shown to depend on the degree of filling of the porous medium and its temperature in a critical manner. A physical mechanism is proposed where the transition of a system of liquid nanoclusters in a confinement into a metastable state in narrow filling and temperature ranges results from the appearance of a potential barrier due to the fluctuations of the collective “multiparticle” interaction of liquid nanoclusters in neighboring pores of different sizes at the shell of a percolation cluster of filled pores. The energy of a metastable state forms a potential relief with numerous maxima and minima in the space of a porous medium. The dispersed liquid volume in a metastable state is calculated with an analytical percolation theory for a ground state with an infinite percolation cluster. The outflow time distribution function of pores is calculated, and a power law is obtained for the decrease in nonwetting liquid volume retained in a porous medium with increasing time. The relaxation of the system under study is a multistage process accompanied by discontinuous equilibrium and overcoming of numerous local maxima of a potential relief. The formation of the metastable state of retained nonwetting liquid results from the nonergodicity properties of a disordered porous medium. The proposed model can describe the detected dependences of dispersed liquid volume on the degree of filling and temperature.

Kinetics of the dispersion transition and nonergodicity of a system consisting of a disordered porous medium and a nonwetting liquid

An approach has been proposed for the description of the dispersion transition of a nonwetting liquid in confinement. This approach describes intrusion and extrusion processes for the ground state of a disordered porous medium, which is characterized by the formation of a fractal percolation cluster. The observed transition of the system of liquid nanoclusters in confinement to a metastable state in a narrow range of degrees of filling and temperatures has been explained by the appearance of a potential barrier owing to fluctuations of the collective "multiparticle interaction" of liquid nanoclusters in neighboring pores of different sizes on the shell of the fractal percolation cluster of filled pores. The energy of the metastable state forms a potential relief in the space of the porous medium with many maxima and minima. The volume of the dispersed liquid in the metastable state has been calculated within the analytical percolation theory for the ground state with the infinite percolation cluster. The extrusion-time distribution function of pores has been calculated. It has been found that the volume of the nonwetting liquid remaining in the porous medium decreases with time according to a power law. Relaxation in the system under study is a multistep process involving discontinuous equilibrium and overcoming of many local maxima of the potential relief. The formation of the metastable state of the trapped nonwetting liquid has been attributed to the nonergodicity of the disordered porous medium. The model reproduces the observed dependence of the volume of the dispersed liquid both on the degree of filling and on the temperature.

Dynamical cluster approximation with continuous lattice self-energy

The dynamical cluster approximation (DCA) is a systematic extension beyond the single site approximation in dynamical mean field theory (DMFT), to include spatially non-local correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in oder to achieve better convergence with cluster size. The new method, which we call DCA$^+$, cures the cluster shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that in the DCA$^+$ algorithm, the self-energy and pseudo-gap temperature $T^*$ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-rage correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA$^+$ algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.

Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster

In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on \(\mathbb{Z }^d\). More precisely, we count \(Z_N\), the number of self-avoiding paths of length \(N\) on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of \(Z_N\), \(\limsup _{N\rightarrow \infty } Z_N^{1/N}\), which we call the connective constant of the dilute lattice. After proving that this connective constant is a.s. non-random, we focus on the two-dimensional case and show that for every percolation parameter \(p\in (1/2,1)\), almost surely, \(Z_N\) grows exponentially slower than its expected value. In other words, we prove that \(\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}\), where the expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on \(\mathbb{Z }^2\), so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.

Random interlacements and amenability

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039–2087] for the special case of ${\mathbb{Z}}^{d}$ (with $d\geq3$). In Sznitman [Ann. of Math. (2) (2010) 171 2039–2087], it was shown that on ${\mathbb{Z}}^{d}$: for any intensity $u>0$, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity $u$ the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of $u$. Finally, we establish the monotonicity of the transition between the “disconnected” and the “connected” phases, providing the uniqueness of the critical value $u_{c}$ where this transition occurs.