Two-dimensional Percolation Research Articles

Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules

We study two-dimensional critical bootstrap percolation models. We establish that a class of these models including all isotropic threshold rules with a convex symmetric neighbourhood, undergoes a sharp metastability transition. This extends previous instances proved for several specific rules. The paper supersedes a draft by Alexander Holroyd and the first author from 2012. While it served a role in the subsequent development of bootstrap percolation universality, we have chosen to adopt a more contemporary viewpoint in its present form.

Logarithmic correlation functions in 2D critical percolation

It is believed that the large-scale geometric properties of two-dimensional critical percolation are described by a logarithmic conformal field theory, but it has been challenging to exhibit concrete examples of logarithmic singularities and to find an explanation and a physical interpretation, in terms of lattice observables, for their appearance. We show that certain percolation correlation functions receive independent contributions from a large number of similar connectivity events happening at different scales. Combined with scale invariance, this leads to logarithmic divergences. We study several logarithmic correlation functions for critical percolation in the bulk and in the presence of a boundary, including the four-point function of the density (spin) field. Our analysis confirms previous findings, provides new explicit calculations and explains, in terms of lattice observables, the physical mechanism that leads to the logarithmic singularities we discover. Although we adopt conformal field theory (CFT) terminology to present our results, the core of our analysis relies on probabilistic arguments and recent rigorous results on the scaling limit of critical percolation and does not assume a priori the existence of a percolation CFT. As a consequence, our results provide strong support for the validity of a CFT description of critical percolation and a step in the direction of a mathematically rigorous formulation of a logarithmic CFT of two-dimensional critical percolation.

Critical Permeability from Resummation

Special calculation methods are presented for critical indices and amplitudes for the permeability of thin wavy channels dependent on the waviness. The effective permeability and wetted perimeter of the two-dimensional random percolating media are considered as well. A special mathematical framework is developed to characterize the dependencies on porosities, critical points, and indices. Various approximation techniques are applied without involving popular lubrication approximation in any sense. In particular, the Borel summation technique is applied to the effective polynomial approximations with or without optimization. Minimal difference and minimal derivative optimal conditions are adapted to calculations of critical indices and amplitudes for the effective permeability of thin wavy channels. Critical indices, amplitudes, and thresholds are obtained for the effective permeability and wetted perimeter of the two-dimensional percolating random media. Closed-form expressions for all porosities, critical points, and indices are calculated from the polynomial approximations for the first time.

Disorder-induced topological quantum phase transitions in multigap Euler semimetals

We study the effect of disorder in systems having a nontrivial Euler class. As these recently proposed multigap topological phases come about by braiding non-Abelian charged band nodes residing between different bands to induce stable pairs within isolated band subspaces, novel properties may be expected. Namely, a modified stability and critical phases under the unbraiding to metals can arise when the disorder preserves the underlying C2T or PT symmetry on average. Employing elaborate numerical computations, we verify the robustness of associated topology by evaluating the changes in the average densities of states and conductivities for different types of disorders. Upon performing a scaling analysis around the corresponding quantum critical points, we retrieve a universality for the localization length exponent of ν=1.4±0.1 for Euler-protected phases, relating to two-dimensional percolation models. We generically find that quenched disorder drives Euler semimetals into critical metallic phases. Finally, we show that magnetic disorder can also induce topological transitions to quantum anomalous Hall plaquettes with local Chern numbers determined by the initial value of the Euler invariant. Published by the American Physical Society 2024

Continuum percolation of two-dimensional adaptive dynamics systems.

The percolation phase transition of a continuum adaptive neuron system with homeostasis is investigated. In order to maintain their average activity at a particular level, each neuron (represented by a disk) varies its connection radius until the sum of overlapping areas with neighboring neurons (representing the overall connection strength in the network) has reached a fixed target area for each neuron. Tuning the two key parameters in the model, i.e., the density defined as the number of neurons (disks) per unit area and the sum of the overlapping area of each disk with its adjacent disks, can drive the system into the critical percolating state. These two parameters are inversely proportional to each other at the critical state, and the critical filling factors are fixed about 0.7157, which is much less than the case of the continuum percolation with uniform disks. It is also confirmed that the critical exponents in this model are the same as the two-dimensional standard lattice percolation. Although the critical state is relatively more sensitive and exhibits long-range spatial correlation, local fluctuations do not propagate in a long-range manner through the system by the adaptive dynamics, which renders the system overall robust against perturbations.

Short-range correlation of stress chains near solid-to-liquid transition in active monolayers.

Using a three-dimensional model of cell monolayers, we study the spatial organization of active stress chains as the monolayer transitions from a solid to a liquid state. The critical exponents that characterize this transition map the isotropic stress percolation onto the two-dimensional random percolation universality class, suggesting short-range stress correlations near this transition. This mapping is achieved via two distinct, independent pathways: (i) cell-cell adhesion and (ii) active traction forces. We unify our findings by linking the nature of this transition to high-stress fluctuations, distinctly linked to each pathway. The results elevate the importance of the transmission of mechanical information in dense active matter and provide a new context for understanding the non-equilibrium statistical physics of phase transition in active systems.

Electrical conductance of two-dimensional random percolating networks based on mixtures of nanowires and nanorings: A mean-field approach along with computer simulation.

We have studied the electrical conductance of two-dimensional (2D) random percolating networks of zero-width metallic nanowires (a mixture of rings and sticks). We took into account the nanowire resistance per unit length and the junction (nanowire-nanowire contact) resistance. Using a mean-field approximation (MFA) approach, we derived the total electrical conductance of these nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo (MC) numerical simulations. The MC simulations were focused on the case when the circumferences of the rings and the lengths of the wires were equal. In this case, the electrical conductance of the network was found to be almost insensitive to the relative proportions of the rings and sticks, provided that the wire resistance and the junction resistance were equal. When the junction resistance dominated over the wire resistance, a linear dependency of the electrical conductance of the network on the proportions of the rings and sticks was observed.

Topological random fractals

The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of topological electronic states to random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from a two-dimensional site percolation process, we demonstrate that these topological random fractals exhibit the hallmarks of topological insulators. Specifically, our large-scale numerical studies reveal that topological random fractals display a robust mobility gap, support quantized conductance and represent a well-defined thermodynamic phase of matter. The finite-size scaling analysis further suggests that the critical properties are not consistent with the expectations of class D systems in two dimensions, hinting to the nontrivial relationship between fractal and integer-dimensional topological states. Our results establish topological random fractals as the most complex systems known to support nontrivial band topology with their distinct unique properties.

Sharp threshold for two-dimensional majority dynamics percolation

In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in\mathbb{R}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical probability parameter $p_c(t)$ with polynomial size window. We then use this result in order to obtain stretched-exponential bounds on the one-arm event probability in the subcritical phase. Our results are based on differential inequalities derived from the OSSS inequality, inspired by the recent developments by Ahlberg, Broman, Griffiths, and Morris and by Duminil-Copin, Raoufi, and Tassion. We also provide analogous results for percolation in the voter model.

Random 2D nanowire networks: Finite-size effect and the effect of busbar/nanowire contact resistance on their electrical conductivity

We have studied the resistance of two-dimensional random percolating networks of zero-width metallic nanowires (rings or sticks). We took into account the nanowire resistance per unit length, the junction (nanowire/nanowire contact) resistance, and the busbar/nanowire contact resistance. Using a mean-field approximation (MFA), we derived the total resistance of the nanoring-based networks as a function of their geometrical and physical parameters. We have proposed a way of accounting for the contribution of the busbar/nanowire contact resistance toward the network resistance. The MFA predictions have been confirmed by our Monte Carlo numerical simulations. Our study evidenced that the busbar/nanowire contact resistance has a significant effect on the electrical conductivity when the junction resistance dominates over the wire resistance.

Invasiveness of a Growth-Migration System in a Two-dimensional Percolation cluster: A Stochastic Mathematical Approach.

We developed a framework based on the software Unstructured Reaction-Diffusion Master Equation (URDME) to address tumor cells' proliferation and migration in a heterogeneous space, herein a 2D percolation cluster. A mitogenic paracrine signaling pathway is utilized phenomenologically to reveal how cells cooperate with one another. We modeled the emerging Allee effect using low seeding density culture (LSDC) assays to fit the model parameters. A Finite time scaling (FTS) function has been formulated to quantitatively analyze invasiveness of a virtual Growth-Migration (GM) system in mimicking the cancer cell growth. Through such simulation, we analyzed the GM dynamics of virtual model in mimicking the growth of BT-474 cancer cell populations in vitro in a 2D percolation cluster and calculated the successful penetration rate (SPR). By analyzing the temporal trajectories of the SPR, we could determine the critical exponents of the critical SPR scaling relation. The SPR transition point ([Formula: see text]), which is a fundamentally different from a conventional percolation transition point, is found to be negatively correlated with the invasiveness of this cancer cell. The [Formula: see text] of the three variations of the virtual GM system distinctly designated by varying paracrine-regulated Allee (PAllee) model phenotypes is 0.3408, 0.3675, and 0.4454, respectively. FTS algorithm thereon may serve as an approach to quantify invasiveness of tumor cells. Through a phenomenological paracrine model, inter-cell cooperation and mutual mitogenic boosting are enabled to elicit the Allee effect in the GM systems. The rationale behind such computationally tunable virtual mechanism can be applied to other circumstances concerning emerging processes.

Dependence of the conductivity of two-dimensional site percolation network on the length-ratio of conducting paths to all bonds: the viewpoint of effective path theory

Previous studies have found that the network conductivity of 2-dimensional disordered nanowire networks (DNNs) scaled linearly with the length-ratio of conducting-paths to all nanowires. To show the universality of this rule, the conducting behavior of a 2-dimensional site percolation problem is studied in this article with the assistance of a Monte Carlo based numerical simulation. It is observed that, as the existence probability of site increases in the 2-dimensional site percolated network, more conducting-paths are formed, and the network becomes more conductive. After correlating the site-percolated lattice to DNNs, the normalized network conductivity is observed to scale linearly with the length-ratio of conducting-paths to all bonds, which could be well described by the linear formula using a slope of 2 and an incept of 0.5. As a result, the length-ratio of conducting-paths could again serve as a basic topological parameter in describing the conducting behavior of 2-dimensional site percolation networks. Such universality enables the definition of an ‘effective path theory’, in which the normalized network conductivity scales linearly with the length-ratio of conducting-paths to all bonds.

Transfer learning of phase transitions in percolation and directed percolation.

The latest advances of statistical physics have shown remarkable performance of machine learning in identifying phase transitions. In this paper, we apply domain adversarial neural network (DANN) based on transfer learning to studying nonequilibrium and equilibrium phase transition models, which are percolation model and directed percolation (DP) model, respectively. With the DANN, only a small fraction of input configurations (two-dimensional images) needs to be labeled, which is automatically chosen, to capture the critical point. To learn the DP model, the method is refined by an iterative procedure in determining the critical point, which is a prerequisite for the data collapse in calculating the critical exponent ν_{⊥}. We then apply the DANN to a two-dimensional site percolation with configurations filtered to include only the largest cluster which may contain the information related to the order parameter. The DANN learning of both models yields reliable results which are comparable to the ones from Monte Carlo simulations. Our study also shows that the DANN can achieve quite high accuracy at much lower cost, compared to the supervised learning.

A random hermitian matrix representation for two-dimensional percolation model‎

In this letter, the random matrix theory representation of a bond-percolation model on square lattice is presented. The behavior of random matrix model can be determined only by its two largest eigenvalues. The second largest eigenvalue sits exactly on the edge of semicircle part of eigenvalue’s distribution and its position is a function of p, on the other hand the first largest eigenvalue is disjointed from other eigenvalues and its distribution is Gaussian. Also the first largest eigenvalue is responsible for scaling properties near criticality. Numerical simulations show power-law divergences emerged from coalescence of two largest eigenvalues near critical point at the thermodynamic limit. In this letter a scaling formalism is presented which describes complete scaling behavior of largest eigenvalue’s fluctuations with a set of scaling exponents in finite-size systems.

Nested closed paths in two-dimensional percolation

For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path (NP) operator and thus a continuous family of one-point functions , where ℓ is the number of independent (i.e., non-overlapping) nested closed paths surrounding the center, k is a path fugacity, and projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling , with L the linear system size, and we determine the exponent X NP as a function of k. On the basis of our numerical results, we conjecture an analytical formula, with k = 2 cos(πϕ), which reproduces the exact results for k = 0, 1 and agrees with the high-precision estimate of for other k values. In addition, we observe that W 2(L) = 1 for site percolation on the triangular lattice with any size L, and we prove this identity for all self-matching lattices.

Two-dimensional continuum percolation models with disks under the generalized Achlioptas process

We investigate the disk percolation phase transitions in continuum models in two dimensions under four kinds of generalized product rule and four kinds of generalized sum rule, respectively. We study the critical behaviors of the largest, second largest clusters and their size ratio to characterize the universality class of percolation transition. Using the finite size scaling analysis and the Monte Carlo simulation, we calculate the critical exponents of the percolation transitions under the above rules. We find that the universality classes of continuum percolation under the generalized product rule, under the generalized sum rule, and the lattice percolation are different.

Universal scaling and criticality of extremes in random matrix theory.

We present a random matrix realization of a two-dimensional percolation model with the occupation probability p. We find that the behavior of the model is governed by the two first extreme eigenvalues. While the second extreme eigenvalue resides on the moving edge of the semicircle bulk distribution with an additional semicircle functionality on p, the first extreme exhibits a disjoint isolated Gaussian statistics which is responsible for the emergence of a rich finite-size scaling and criticality. Our extensive numerical simulations along with analytical arguments unravel the power-law divergences due to the coalescence of the first two extreme eigenvalues in the thermodynamic limit. We develop a scaling law that provides a universal framework in terms of a set of scaling exponents uncovering the full finite-size scaling behavior of the extreme eigenvalue's fluctuation. Our study may provide a simple practical approach to capture the criticality in complex systems and their inverse problems with a possible extension to the interacting systems.

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.

Chemical Subdiffusivity of Critical 2D Percolation

We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (Probab Theory Related Fields 73(3): 369–394 1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multiscale way of covering the graph so that “deep patches” have “thin backbones.” Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents \(\eta _2> \eta _1 > 0\): For d-dimensional models, the mean chemical displacement after T steps of random walk scales asymptotically slower than \(T^{1/\beta }\), whenever $$\begin{aligned} \beta < 2 + \frac{\eta _2-\eta _1}{d-\eta _1}\,. \end{aligned}$$Using the conjectured values of \(\eta _2 = \eta _1 + 1/4\) and \(\eta _1 = 5/48\) for 2D lattices, the latter quantity is \(2+12/91\).

Fluctuations of transverse increments in two-dimensional first passage percolation

We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d. nonnegative, continuous, and have a finite moment generating function in a neighborhood of 0. We derive consequences about transverse increments of passage times, assuming the model satisfies certain properties. Approximately, the assumed properties are the following: We assume that the standard deviation of the passage time on scale r is of some order σ(r), and σ(r),r>0 grows approximately as a power of r. Also, the tails of the passage time distributions for distance r satisfy an exponential bound on a scale σ(r) uniformly over r. In addition, the boundary of the limit shape in a neighborhood of some fixed direction θ has a uniform quadratic curvature. By transverse increment we mean the difference between passage times from the origin to a pair of points which are approximately at the direction θ and the direction between the pair of points is the direction of the tangent to the boundary of the limit shape at the direction θ. The main consequence derived is the following. If σ(r) varies as rχ for some χ>0, and ξ is such that χ=2ξ−1, then the fluctuation of the transverse increment of passage time between a pair of points situated at distance r from each other is of the order of rχ∕ξ.