Critical Percolation Clusters Research Articles

Iterative site percolation on triangular lattice

The site percolation on the triangular lattice stands out as one of the few exactly solved statistical systems. By initially configuring critical percolation clusters of this model and randomly reassigning the color of each percolation cluster, we obtain coarse-grained configurations by merging adjacent clusters that share the same color. It is shown that the process can be infinitely iterated in the infinite-lattice limit, leading to an iterative site percolation model. We conjecture from the self-matching argument that percolation clusters remain fractal for any finite generation, which can even take any positive real number by a generalized process. Extensive simulations are performed, and, from the generation-dependent fractal dimension, a continuous family of previously unknown universalities is revealed. Published by the American Physical Society 2024

Decomposing the percolation backbone reveals novel scaling laws of the current distribution

The distribution of currents on critical percolation clusters is the fundamental quantity describing the transport properties of weakly connected systems. Nevertheless, its finite-size extrapolation is still one of the outstanding open questions concerning disordered media. By hierarchically decomposing the 3-connected components of the backbone, we disclose that the current distribution is determined from two distributions, namely, the one corresponding to the number of bonds in each level and another one corresponding to the factors by which the current is reduced, when going from one level to the next. The first distribution follows a finite-size scaling, while the second is a power law with an exponent consistent with 3/4 in two dimensions. The standard hierarchical model for the backbone is too simple to reproduce this complex scenario. Our new decomposition method of the backbone also allows to calculate much smaller currents than before, attaining a precision of 10−35 and systems of size L = 81922. Moreover, our method is not restricted to electric currents on critical percolation clusters but could also be applied to other transport problems on sparse graphs including fluid flow and car traffic.

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder: II. Rotated lattice

This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the 45∘ rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allows us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder, respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov–Stroganov point, while the effective tilt is introduced via the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin–Stroganov–Zagier’s solution of Baxter’s T–Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.

Data-driven discovery of the governing equationsfor transport in heterogeneous media by symbolic regression and stochastic optimization.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equationsof continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equationsfor flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equationsfor flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equationfor anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

Universal fluctuations and ergodicity of generalized diffusivity on critical percolation clusters

Despite a long history and a clear overall understanding of properties of random walks on an incipient infinite cluster in percolation, some important information on it seems to be missing in the literature. In the present work, we revisit the problem by performing massive numerical simulations for (sub)diffusion of particles on such clusters. Thus, we discuss the shape of the probability density function of particles’ displacements, and the way it converges to its long-time limiting scaling form. Moreover, we discuss the properties of the mean squared displacement (MSD) of a particle diffusing on the infinite cluster at criticality. This one is known not to be self-averaging. We show that the fluctuations of the MSD in different realizations of the cluster are universal, and discuss the properties of the distribution of these fluctuations. These strong fluctuations coexist with the ergodicity of subdiffusive behavior in the time domain. The dependence of the relative strength of fluctuations in time-averaged MSD on the total trajectory length (total simulation time) is characteristic for diffusion in a percolation system and can be used as an additional test to distinguish this process with disorder-induced memory from processes with otherwise similar behavior, like fractional Brownian motion with the same value of the Hurst exponent.

An estimate for the radial chemical distance in [formula omitted] critical percolation clusters

We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length 2n, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path γ from the origin to distance n that consists of “three-arm” points, and whose volume can thus be estimated by O(n2π3(n)). Here, π3(n) is the “three-arm probability” that the origin is connected to distance n by three arms, two open and one dual-closed. We then develop estimates for the existence of shortcuts around an edge e in the box, conditional on {e∈γ}, to obtain a bound of the form O(n2−δπ3(n)) for some δ>0.

Global lower mass-bound for critical configuration models in the heavy-tailed regime

We establish the global lower mass-bound property for the largest connected components in the critical window for the configuration model when the degree distribution has an infinite third moment. The scaling limit of the critical percolation clusters, viewed as measured metric spaces, was established in our prior work with respect to the Gromov-weak topology. Our result extends those scaling limit results to the stronger Gromov-Hausdorff-Prokhorov topology under slightly stronger assumptions on the degree distribution. This implies the distributional convergence of global functionals such as the diameter of the largest critical components. Further, our result gives a sufficient condition for compactness of the random metric spaces that arise as scaling limits of critical clusters in the heavy-tailed regime.

Critical exponents and universal excess cluster number of percolation in four and five dimensions

We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the leaf-free and the bridge-free clusters using the breadth-first search algorithm. From the geometric properties of these clusters, we determine a set of four critical exponents, including the thermal exponent yt≡1∕ν, the fractal dimension df, the backbone exponent dB and the shortest-path exponent dmin. We also obtain an estimate of the excess cluster number b which is a universal quantity related to the finite-size scaling of the total number of clusters. The results are yt=1.461(5), df=3.0446(7), dB=1.9844(11), dmin=1.6042(5), b=0.62(1) for 4D; and yt=1.743(10), df=3.5260(14), dB=2.0226(27), dmin=1.8137(16), b=0.62(2) for 5D. The values of the critical exponents are compatible with or improving over the existing estimates, and those of the excess cluster number b have not been reported before. Together with the existing values in other spatial dimensions d, the d-dependent behavior of the critical exponents is obtained, and a local maximum of dB is observed near d≈5. It is suggested that, as expected, critical percolation clusters become more and more dendritic as d increases.

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder

We obtain exact densities of contractible and non-contractible loops in the O(1) model on a strip of the square lattice rolled into an infinite cylinder of finite even circumference L. They are also equal to the densities of critical percolation clusters on 45 degree rotated square lattice rolled into a cylinder, which do not or do wrap around the cylinder respectively. The results are presented as explicit rational functions of L taking rational values for any even L. Their asymptotic expansions in the large L limit have irrational coefficients reproducing the earlier results in the leading orders. The solution is based on a mapping to the six-vertex model and the use of technique of Baxter’s T–Q equation.

Time increasing rates of infiltration and reaction in porous media at the percolation thresholds.

The infiltration of a solute in a fractal porous medium is usually anomalous, but chemical reactions of the solute and that material may increase the porosity and affect the evolution of the infiltration. We study this problem in two- and three-dimensional lattices with randomly distributed porous sites at the critical percolation thresholds and with a border in contact with a reservoir of an aggressive solute. The solute infiltrates that medium by diffusion and the reactions with the impermeable sites produce new porous sites with a probability r, which is proportional to the ratio of reaction and diffusion rates at the scale of a lattice site. Numerical simulations for r≪1 show initial subdiffusive scaling and long time Fickean scaling of the infiltrated volumes or areas, but with an intermediate regime with time increasing rates of infiltration and reaction. The anomalous exponent of the initial regime agrees with a relation previously applied to infinitely ramified fractals. We develop a scaling approach that explains the subsequent time increase of the infiltration rate, the dependence of this rate on r, and the crossover to the Fickean regime. The exponents of the scaling relations depend on the fractal dimensions of the critical percolation clusters and on the dimensions of random walks in those clusters. The time increase of the reaction rate is also justified by that reasoning. As r decreases, there is an increase in the number of time decades of the intermediate regime, which suggests that the time increasing rates are more likely to be observed is slowly reacting systems.

Worm-algorithm-type simulation of the quantum transverse-field Ising model

We apply a worm algorithm to simulate the quantum transverse-field Ising model in a path-integral representation of which the expansion basis is taken as the spin component along the external-field direction. In such a representation, a configuration can be regarded as a set of non-intersecting loops constructed by "kinks" for pairwise interactions and spin-down (or -up) imaginary-time segments. The wrapping probability for spin-down loops, a dimensionless quantity characterizing the loop topology on a torus, is observed to exhibit small finite-size corrections and yields a high-precision critical point in two dimensions (2D) as $h_c \! =\! 3.044\, 330(6)$, significantly improving over the existing results and nearly excluding the best one $h_c \! =\! 3.044\, 38 (2)$. At criticality, the fractal dimensions of the loops are estimated as $d_{\ell \downarrow} (1{\rm D}) \! = \! 1.37(1) \! \approx \! 11/8 $ and $d_{\ell \downarrow} (2{\rm D}) \! = \! 1.75 (3)$, consistent with those for the classical 2D and 3D O(1) loop model, respectively. An interesting feature is that in one dimension (1D), both the spin-down and -up loops display the critical behavior in the whole disordered phase ($ 0 \! \leq \! h \! < \! h_c$), having a fractal dimension $d_{\ell} \! = \! 1.750 (7)$ that is consistent with the hull dimension $d_{\rm H} \! = \! 7/4$ for critical 2D percolation clusters. The current worm algorithm can be applied to simulate other quantum systems like hard-core boson models with pairing interactions.

Critical percolation and the incipient infinite cluster on Galton-Watson trees

We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.

The geometry of a critical percolation cluster on the UIPT

Nous examinons le modèle de percolation de Bernoulli par sites critique sur la triangulation infinie uniforme du plan. Nous étudions les queues de distribution du temps d’exploration, du périmètre et du volume de l’enveloppe d’une composante connexe. Les exposants obtenus diffèrent d’un facteur $2$ de ceux calculés auparavant par Angel et Curien [Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 405–431] dans le cas de la percolation critique par site sur la triangulation uniforme du demi-plan.

Critical percolation clusters in seven dimensions and on a complete graph.

We study critical bond percolation on a seven-dimensional hypercubic lattice with periodic boundary conditions (7D) and on the complete graph (CG) of finite volume (number of vertices) V. We numerically confirm that for both cases, the critical number density n(s,V) of clusters of size s obeys a scaling form n(s,V)∼s^{-τ}n[over ̃](s/V^{d_{f}^{*}}) with identical volume fractal dimension d_{f}^{*}=2/3 and exponent τ=1+1/d_{f}^{*}=5/2. We then classify occupied bonds into bridge bonds, which includes branch and junction bonds, and nonbridge bonds; a bridge bond is a branch bond if and only if its deletion produces at least one tree. Deleting branch bonds from percolation configurations produces leaf-free configurations, whereas deleting all bridge bonds leads to bridge-free configurations composed of blobs. It is shown that the fraction of nonbridge (biconnected) bonds vanishes, ρ_{n,CG}→0, for large CGs, but converges to a finite value, ρ_{n,7D}=0.0061931(7), for the 7D hypercube. Further, we observe that while the bridge-free dimension d_{bf}^{*}=1/3 holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: d_{lf,7D}^{*}=0.669(9)≈2/3 and d_{lf,CG}^{*}=0.3337(17)≈1/3. On the CG and in 7D, the whole, leaf-free, and bridge-free clusters all have the shortest-path volume fractal dimension d_{min}^{*}≈1/3, characterizing their graph diameters. We also study the behavior of the number and the size distribution of leaf-free and bridge-free clusters. For the number of clusters, we numerically find the number of leaf-free and bridge-free clusters on the CG scale as ∼lnV, while for 7D they scale as ∼V. For the size distribution, we find the behavior on the CG is governed by a modified Fisher exponent τ^{'}=1, while for leaf-free clusters in 7D, it is governed by Fisher exponent τ=5/2. The size distribution of bridge-free clusters in 7D displays two-scaling behavior with exponents τ=4 and τ^{'}=1. The probability distribution P(C_{1},V)dC_{1} of the largest cluster of size C_{1} for whole percolation configurations is observed to follow a single-variable function P[over ¯](x)dx, with x≡C_{1}/V^{d_{f}^{*}} for both CG and 7D. Up to a rescaling factor for the variable x, the probability functions for CG and 7D collapse on top of each other within the entire range of x. The analytical expressions in the x→0 and x→∞ limits are further confirmed. Our work demonstrates that the geometric structure of high-dimensional percolation clusters cannot be fully accounted for by their complete-graph counterparts.

Topological Hausdorff dimension and geodesic metric of critical percolation cluster in two dimensions

In this work, we prove that the topological Hausdorff dimension of critical percolation cluster (CPC) in two dimensions is equal to DtH=Drb+1=7/4, where Drb is the Hausdorff dimension of the set of red bonds. Hence, the CPC is infinitely ramified. We also argue that the mapping from the Euclidean metric to the geodesic metric on the CPC is governed by the Hausdorff dimension of the cluster skeleton Dsc=DH/dℓ>dmin, where DH, dℓ, and dmin are the Hausdorff and the connectivity (chemical) dimensions of the CPC and the fractal dimension of the minimum path, respectively. Then we introduce the notion of the topological connectivity dimension dtℓ. This allows us to establish the exact upper and lower bounds for the connectivity dimension dℓ of the CPC in d=2. The upper and lower bounds for some other dimension numbers were established using the relations between dimension numbers. Narrow ranges defined by these bounds are much smaller than the error bars of numerical estimates reported in literature. Accordingly, the exact values of some dimension numbers are conjectured.

Exact enumeration of self-avoiding walks on critical percolation clusters in 2–7 dimensions

We study self-avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method, which can handle walks of several thousand steps. We had previously presented results for the two- and three-dimensional cases; here we take a wider perspective and vary the system’s dimensions up to D = 7, beyond the supposed upper critical dimension of . These results may serve to check analytical predictions and help understand how the medium’s fractal structure impacts on the walks’ scaling behavior.

Analysis of critical percolation clusters of mixtures of conducting and dielectric polymers

Analysis of critical percolation clusters of mixtures of conducting and dielectric polymers

The expected number of critical percolation clusters intersecting a line segment

We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrict to the upper halfplane, or $\mathbb{C}$, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that $E_{\mathbb{H}}(n) = An + \frac{\sqrt{3}}{4\pi}\log(n) + o(\log(n))$, where $A$ is some constant. Recently Kov\'{a}cs, Igl\'{o}i and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for $E_{\mathbb{C}}(n)$ with the constant $\frac{\sqrt{3}}{4\pi}$ replaced by $\frac{5\sqrt{3}}{32\pi}$. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of $E_{\mathbb{H}}(n)$ above, and a rigorous upper bound for the prefactor of the logarithm in the formula of $E_{\mathbb{C}}(n)$.

Recursive percolation.

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and find compelling numerical evidence that it can be repeated recursively any number n of generations. In two dimensions, we determine the percolation thresholds up to n=5. The corresponding critical clusters become more and more compact as n increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any n. This family of exponents differs from previously known universality classes, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.

Transition from compact to porous films in deposition with temperature-activated diffusion.

We study a thin-film growth model with temperature activated diffusion of adsorbed particles, allowing for the formation of overhangs and pores, but without detachment of adatoms or clusters from the deposit. Simulations in one-dimensional substrates are performed for several values of the diffusion-to-deposition ratio R of adatoms with a single bond and of the detachment probability ε per additional nearest neighbor, respectively, with activation energies are E(s) and E(b). If R and ε independently vary, regimes of low and high porosity are separated at 0.075≤ε(c)≤0.09, with vanishingly small porosity below that point and finite porosity for larger ε. Alternatively, for fixed values of E(s) and E(b) and varying temperature, the porosity has a minimum at T(c), and a nontrivial regime in which it increases with temperature is observed above that point. This is related to the large mobility of adatoms, resembling features of equilibrium surface roughening. In this high-temperature region, the deposit has the structure of a critical percolation cluster due to the nondesorption. The pores are regions enclosed by blobs of the corresponding percolating backbone, thus the distribution of pore size s is expected to scale as s(-τ̃) with τ̃≈1.45, in reasonable agreement with numerical estimates. Roughening of the outer interface of the deposits suggests Villain-Lai-Das Sarma scaling below the transition. Above the transition, the roughness exponent α≈0.35 is consistent with the percolation backbone structure via the relation α=2-d(B), where d(B) is the backbone fractal dimension.