Percolation with long-range correlated disorder

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.

Emergence of rigidity percolation in flowing granular systems

Jammed granular media and glasses exhibit spatial long-range correlations as a result of mechanical equilibrium. However, the existence of such correlations in the flowing matter, where the mechanical equilibrium is unattainable, has remained elusive. Here, we investigate this problem in the context of the percolation of interparticle forces in flowing granular media. We find that the flow rate introduces an effective long-range correlation, which plays the role of a relevant perturbation giving rise to a spectrum of varying exponents on a critical line as a function of the flow rate. Our numerical simulations along with analytical arguments predict a crossover flow rate [Formula: see text] below which the effect of induced disorder is weak and the universality of the force chain structure is shown to be given by the standard rigidity percolation. We also find a power-law behavior for the critical exponents with the flow rate [Formula: see text].

Aftermath epidemics: Percolation on the sites visited by generalized random walks

We study percolation on the sites of a finite lattice visited by a generalized random walk of finite length with periodic boundary conditions. More precisely, consider Levy flights and walks with finite jumps of length >1 [like Knight's move random walks (RWs) in two dimensions and generalized Knight's move RWs in 3D]. In these walks, the visited sites do not form (as in ordinary RWs) a single connected cluster, and thus percolation on them is nontrivial. The model essentially mimics the spreading of an epidemic in a population weakened by the passage of some devastating agent-like diseases in the wake of a passing army or of a hurricane. Using the density of visited sites (or the number of steps in the walk) as a control parameter, we find a true continuous percolation transition in all cases except for the 2D Knight's move RWs and Levy flights with Levy parameter σ≥2. For 3D generalized Knight's move RWs, the model is in the universality class of pacman percolation, and all critical exponents seem to be simple rationals, in particular, β=1. For 2D Levy flights with 0<σ<2, scale invariance is broken even at the critical point, which leads at least to very large corrections in finite-size scaling, and even very large simulations were unable to unambiguously determine the critical exponents.

Breaking universality in random sequential adsorption on a square lattice with long-range correlated defects

Jamming and percolation transitions in the standard random sequential adsorption of particles on regular lattices are characterized by a universal set of critical exponents. The universality class is preserved even in the presence of randomly distributed defective sites that are forbidden for particle deposition. However, using large-scale Monte Carlo simulations by depositing dimers on the square lattice and employing finite-size scaling, we provide evidence that the system does not exhibit such well-known universal features when the defects have spatial long-range (power-law) correlations. The critical exponents ν_{j} and ν associated with the jamming and percolation transitions, respectively, are found to be nonuniversal for strong spatial correlations and approach systematically their own universal values as the correlation strength is decreased. More crucially, we have found a difference in the values of the percolation correlation length exponent ν for a small but finite density of defects with strong spatial correlations. Furthermore, for a fixed defect density, it is found that the percolation threshold of the system, at which the largest cluster of absorbed dimers first establishes the global connectivity, gets reduced with increasing the strength of the spatial correlation.

Level Set Percolation in the Two-Dimensional Gaussian Free Field

The nature of level set percolation in the two-dimensional Gaussian free field has been an elusive question. Using a loop-model mapping, we show that there is a nontrivial percolation transition and characterize the critical point. In particular, the correlation length diverges exponentially, and the critical clusters are "logarithmic fractals," whose area scales with the linear size as A∼L^{2}/sqrt[lnL]. The two-point connectivity also decays as the log of the distance. We corroborate our theory by numerical simulations. Possible conformal field theory interpretations are discussed.

Emergent solidity of amorphous materials as a consequence of mechanical self-organisation

Amorphous solids have peculiar properties distinct from crystals. One of the most fundamental mysteries is the emergence of solidity in such nonequilibrium, disordered state without the protection by long-range translational order. A jammed system at zero temperature, although marginally stable, has solidity stemming from the space-spanning force network, which gives rise to the long-range stress correlation. Here, we show that such nonlocal correlation already appears at the nonequilibrium glass transition upon cooling. This is surprising since we also find that the system suffers from giant anharmonic fluctuations originated from the fractal-like potential energy landscape. We reveal that it is the percolation of the force-bearing network that allows long-range stress transmission even under such circumstance. Thus, the emergent solidity of amorphous materials is a consequence of nontrivial self-organisation of the disordered mechanical architecture. Our findings point to the significance of understanding amorphous solids and nonequilibrium glass transition from a mechanical perspective.

Fracture dynamics of correlated percolation on ionomer networks

This article presents a random network model to the study fracture dynamics on a scaffold of charged and elastic ionomer bundles that constitute the stable skeleton of a polymer electrolyte membrane. The swelling pressure upon water uptake by this system creates the internal stress under which ionomer bundles undergo breakage. Depending on the local stress and the strength of bundle-to-bundle correlations, different fracture regimes can be observed. We use kinetic Monte Carlo simulations to study these dynamics. The breakage of individual bundles is described with an exponential breakdown rule and the stress transfer from failed to intact bundles is assumed to exhibit a power-law-type spatial decay. A central property considered in the analysis is the frequency distribution of percolation thresholds, which is employed to analyze fracture regimes as a function of the stress and the effective range of stress transfer. Based on this distribution, we introduce an order parameter for the transition from random breakage to crack growth regimes. Moreover, as a practically important outcome, the time to fracture is analyzed as a descriptor for the lifetime of polymer electrolyte membranes.

Percolation framework of the Earth's topography

Self-similarity and long-range correlations are the remarkable features of the Earth's surface topography. Here we develop an approach based on percolation theory to study the geometrical features of Earth. Our analysis is based on high-resolution, 1 arc min, ETOPO1 global relief records. We find some evidence for abrupt transitions that occurred during the evolution of the Earth's relief network, indicative of a continental/cluster aggregation. We apply finite-size-scaling analysis based on a coarse-graining procedure to show that the observed transition is most likely discontinuous. Furthermore, we study the percolation on two-dimensional fractional Brownian motion surfaces with Hurst exponent H as a model of long-range correlated topography, which suggests that the long-range correlations may play a key role in the observed discontinuity on Earth. Our framework presented here provides a theoretical model to better understand the geometrical phase transition on Earth, and it also identifies the critical nodes that will be more exposed to global climate change in the Earth's relief network.

Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

Motivated by the fact that many physical landscapes are characterized by long-range height-height correlations that are quantified by the Hurst exponent H, we investigate the statistical properties of the iso-height lines of correlated surfaces in the framework of Schramm-Loewner evolution (SLE). We show numerically that in the continuum limit the external perimeter of a percolating cluster of correlated surfaces with H ∈ [-1, 0] is statistically equivalent to SLE curves. Our results suggest that the external perimeter also retains the Markovian properties, confirmed by the absence of time correlations in the driving function and the fact that the latter is Gaussian distributed for any specific time. We also confirm that for all H the variance of the winding angle grows logarithmically with size.

Viscoelasticity of dense suspensions of thermosensitive microgel mixtures undergoing colloidal gelation

Dense suspensions of temperature (T)-sensitive poly(N-isopropyl acrylamide) (N) and poly(N-isopropyl methacrylamide) (NM) microgel mixtures with different volume transition temperatures (T and T, respectively; T < T) exhibit a characteristic T-dependent viscoelasticity due to T-induced changes in the type of interparticle interaction as well as the volume fraction of each gel. In the range of T < T, where the swollen microgels with repulsive interparticle interactions are densely packed, the equilibrium modulus (G) decreases upon heating due entirely to the packing effect, i.e., a reduction in the total volume fraction of the microgels (φ). At T > T where the attractive interparticle interactions between dehydrated and hydrophobic microgels emerge, the suspensions show solid-like elastic properties due to the network-like flocculation of the shrunken microgels (colloidal gelation), even when φ becomes considerably lower than the threshold for randomly close packing. The T-dependence of G shows a minimum at a characteristic temperature (T_B; T_B > T) due to the competition between the repulsive interparticle interactions from the packing effect and electrostatic force, and the attractive interactions from the hydrophobicity. The T_B in N/NM mixture suspensions shifts to a higher value with a decrease in N content in the mixtures (X_N), accompanied by a discontinuous-like change at a specific value of X_N (X_N*). The T_B at every value of X_N agrees approximately with the temperature where the total volume fraction of the attractive hydrophobic microgels is 0.3 regardless of microgel type (N or NM). The discontinuous-like variation in T_B at X_N* reflects the change in the network-like flocculation particles, from only attractive N microgels in the high X_N regime, to the attractive N and NM microgel mixtures in the moderate X_N regime. The requirement of the repulsive electrostatic force with an appropriate strength for the stability of the network-like flocculation is also demonstrated using the PNIPAM-co-fumaric acid (NF) microgel suspensions at various pH.

Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength. The correlations are generated using a discrete version of the Fourier filtering method. We consider two different metrics to set the length scales over which the correlations decay, showing that the percolation thresholds are highly sensitive to such system details. By contrast, we verify that the fractal dimension d_{f} is a universal quantity and unaffected by the choice of metric. We also show that for weak correlations, its value coincides with that for the uncorrelated system. In two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength, approaching d_{f}→2. The onset of this change does not seem to be determined by the extended Harris criterion.

Colored percolation

A model called "colored percolation" has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability p and are then colored by one of the n distinct colors using uniform probability q=1/n. Denoting different colors by the letters of the Roman alphabet, we have studied different versions of the model like AB,ABC,ABCD,ABCDE,... etc. Here, only those lattice bonds having two different colored atoms at the ends are defined as connected. The percolation threshold p_{c}(n) asymptotically converges to its limiting value of p_{c} as 1/n. The model has been generalized by introducing a preference towards a subset of colors when m out of n colors are selected with probability q/m each and the rest of the colors are selected with probability (1-q)/(n-m). It has been observed that p_{c}(q,m) depends nontrivially on q and has a minimum at q_{min}=m/n. In another generalization the fractions of bonds between similarly and dissimilarly colored atoms have been treated as independent parameters. Phase diagrams in this parameter space have been drawn exhibiting percolating and nonpercolating phases.

The influence of statistical properties of Fourier coefficients on random Gaussian surfaces

Many examples of natural systems can be described by random Gaussian surfaces. Much can be learned by analyzing the Fourier expansion of the surfaces, from which it is possible to determine the corresponding Hurst exponent and consequently establish the presence of scale invariance. We show that this symmetry is not affected by the distribution of the modulus of the Fourier coefficients. Furthermore, we investigate the role of the Fourier phases of random surfaces. In particular, we show how the surface is affected by a non-uniform distribution of phases.

Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube?

A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here. Combining numerical simulations and rigorous results, we find off-critical scale-free behavior and a continuous transition at a critical density of holes that significantly differs from classical percolation.

Fluid leakage near the percolation threshold

Percolation is a concept widely used in many fields of research and refers to the propagation of substances through porous media (e.g., coffee filtering), or the behaviour of complex networks (e.g., spreading of diseases). Percolation theory asserts that most percolative processes are universal, that is, the emergent powerlaws only depend on the general, statistical features of the macroscopic system, but not on specific details of the random realisation. In contrast, our computer simulations of the leakage through a seal--applying common assumptions of elasticity, contact mechanics, and fluid dynamics--show that the critical behaviour (how the flow ceases near the sealing point) solely depends on the microscopic details of the last constriction. It appears fundamentally impossible to accurately predict from statistical properties of the surfaces alone how strongly we have to tighten a water tap to make it stop dripping and also how it starts dripping once we loosen it again.

Morphological study of elastic-plastic-brittle transitions in disordered media

We use a spring lattice model with springs following a bilinear elastoplastic-brittle constitutive behavior with spatial disorder in the yield and failure thresholds to study patterns of plasticity and damage evolution. The elastic-perfectly plastic transition is observed to follow percolation scaling with the correlation length critical exponent ν≈1.59, implying the universality class corresponding to the long-range correlated percolation. A quantitative analysis of the plastic strain accumulation reveals a dipolar anisotropy (for antiplane loading) which vanishes with increasing hardening modulus. A parametric study with hardening modulus and ductility controlled through the spring level constitutive response demonstrates a wide spectrum of behaviors with varying degree of coupling between plasticity and damage evolution.

Shortest path and Schramm-Loewner evolution

We numerically show that the statistical properties of the shortest path on critical percolation clusters are consistent with the ones predicted for Schramm-Loewner evolution (SLE) curves for κ = 1.04 ± 0.02. The shortest path results from a global optimization process. To identify it, one needs to explore an entire area. Establishing a relation with SLE permits to generate curves statistically equivalent to the shortest path from a Brownian motion. We numerically analyze the winding angle, the left passage probability, and the driving function of the shortest path and compare them to the distributions predicted for SLE curves with the same fractal dimension. The consistency with SLE opens the possibility of using a solid theoretical framework to describe the shortest path and it raises relevant questions regarding conformal invariance and domain Markov properties, which we also discuss.

Retention capacity of correlated surfaces

We extend the water retention model [C. L. Knecht et al., Phys. Rev. Lett. 108, 045703 (2012)] to correlated random surfaces. We find that the retention capacity of discrete random landscapes is strongly affected by spatial correlations among the heights. This phenomenon is related to the emergence of power-law scaling in the lake volume distribution. We also solve the uncorrelated case exactly for a small lattice and present bounds on the retention of uncorrelated landscapes.