Bond and site percolation in three dimensions

Theory of thermoreversible gelation and anomalous concentration fluctuations in polyzwitterion solutions

We present a theoretical framework to investigate thermoreversible phase transitions within polyzwitterion systems, encompassing macrophase separations (MPS) and gelation. In addition, we explore concentration fluctuations near critical points associated with MPS, as well as tricritical and bicritical points at the intersection of MPS and gelation. By utilizing mean-field percolation theory and field theory formalism, we derive the Landau free energy in terms of polyzwitterion concentration with fixed dipole strengths and other experimental variables, such as temperatures and salt concentrations. As the temperature decreases, the dipoles can form cross-links, resulting in polyzwitterion associations. The associations can grow to a gel network and enhance the propensity for MPS, including liquid-liquid, liquid-gel, and gel-gel phase separations. Remarkably, the associations also impact critical behaviors. Using the renormalization group technique, we find that the critical exponents of the polyzwitterion concentration correlation functions significantly deviate from those in the Ising universality class due to the presence of polyzwitterion associations, leading to crossover critical behaviors.

Simulation study on the effect of polydisperse nanoparticles on polymer diffusion in crowded environments

The diffusion of polymer chains in a crowded environment with large and small immobile, attractive nanoparticles (NPs) is studied using Langevin dynamics simulations. For orderly distributed NPs on the simple cubic lattice, our results show that the diffusion of polymer chains is dependent on the NP-NP distance or lattice distance d. At low d where NPs are placed closely, subdiffusion occurs at a sufficiently high polydispersity of NPs, PD. Both the apparent diffusion coefficient and subdiffusion exponent of polymer chains decrease with increasing PD, attributed to the adsorption of polymers on NP clusters formed by larger NPs. At large d, normal diffusion is always observed, and the diffusion coefficient increases with increasing PD. The reason is that, at high PD, the difference between single large NP adsorption and double large NP adsorption is reduced, which increases the exchange of a polymer between the two adsorption states. Finally, the impact of size polydispersity of NPs on the diffusion of polymer chains in a crowded environment with randomly distributed NPs is also investigated. The results show that the position disorder of NPs enhances the subdiffusion of the system.

Site percolation in distorted square and simple cubic lattices with flexible number of neighbors

This paper exhibits a Monte Carlo study on site percolation using the Newmann-Ziff algorithm in distorted square and simple cubic lattices where each site is allowed to be directly linked with any other site if the Euclidean separation between the pair is at most a certain distance d, called the connection threshold. Distorted lattices are formed from regular lattices by a random but controlled dislocation of the sites with the help of a parameter α, called the distortion parameter. The distinctive feature of this study is the relaxation of the restriction of forming bonds with only the nearest neighbors. Owing to this flexibility and the intricate interplay between the two parameters α and d, the site percolation threshold may either increase or decrease with distortion. The dependence of the percolation threshold on the average degree of a site has been explored to show that the obtained results are consistent with those on percolation in regular lattices with an extended neighborhood and continuum percolation.

Possible new phase transition in the 3D Ising model associated with boundary percolation

In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponentν∼1.3is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponentα∼-1.9. The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two differentνexponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.

Percolation through voids around toroidal inclusions

In the case of media comprised of impermeable particles, fluid flows through voids around impenetrable grains. For sufficiently low concentrations of the latter, spaces around grains join to allow transport on macroscopic scales, whereas greater impenetrable inclusion densities disrupt void networks and block macroscopic fluid flow. A critical grain concentration ρ_{c} marks the percolation transition or phase boundary separating these two regimes. With a dynamical infiltration technique in which virtual tracer particles explore void spaces, we calculate critical grain concentrations for randomly placed interpenetrating impermeable toroidal inclusions; the latter consist of surfaces of revolution with circular and square cross sections. In this manner, we study continuum percolation transitions involving nonconvex grains. As the radius of revolution increases relative to the length scale of the torus cross section, the tori develop a central hole, a topological transition accompanied by a cusp in the critical porosity fraction for percolation. With a further increase in the radius of revolution, as constituent grains become more ringlike in appearance, we find that the critical porosity fraction converges to that of high-aspect-ratio cylindrical counterparts only for randomly oriented grains.

Percolation in a simple cubic lattice with distortion

Site percolation in a distorted simple cubic lattice is characterized numerically employing the Newman-Ziff algorithm. Distortion is administered in the lattice by systematically and randomly dislocating its sites from their regular positions. The amount of distortion is tunable by a parameter called the distortion parameter. In this model, two occupied neighboring sites are considered connected only if the distance between them is less than a predefined value called the connection threshold. It is observed that the percolation threshold always increases with distortion if the connection threshold is equal to or greater than the lattice constant of the regular lattice. On the other hand, if the connection threshold is less than the lattice constant, the percolation threshold first decreases and then increases steadily as distortion is increased. It is shown that the variation of the percolation threshold can be well explained by the change in the fraction of occupied bonds with distortion. The values of the relevant critical exponents of the transition strongly indicate that percolation in regular and distorted simple cubic lattices belong to the same universality class. It is also demonstrated that this model is intrinsically distinct from the site-bond percolation model.

K-selective percolation: A simple model leading to a rich repertoire of phase transitions

We propose a K-selective percolation process as a model for iterative removals of nodes with a specific intermediate degree in complex networks. In the model, a random node with degree K is deactivated one by one until no more nodes with degree K remain. The non-monotonic response of the giant component size on various synthetic and real-world networks implies a conclusion that a network can be more robust against such a selective attack by removing further edges. From a theoretical perspective, the K-selective percolation process exhibits a rich repertoire of phase transitions, including double transitions of hybrid and continuous, as well as reentrant transitions. Notably, we observe a tricritical-like point on Erdős-Rényi networks. We also examine a discontinuous transition with unusual order parameter fluctuation and distribution on simple cubic lattices, which does not appear in other percolation models with cascade processes. Finally, we perform finite-size scaling analysis to obtain critical exponents on various transition points, including those exotic ones.

Percolation on Lieb lattices

We study site- and bond-percolation on a class of lattices referred to as Lieb lattices. In two dimensions the Lieb lattice (LL) is also known as the decorated square lattice, or as the CuO_{2} lattice; in three dimensions it can be generalized to a layered Lieb lattice or to a perovskite lattice. Emergent electronic phenomena, such as topological states and ferrimagnetism, have been predicted to occur in these systems, which may be realized in optical lattices as well as in solid state. Since the study of the interplay between quantum fluctuations and disorder in these systems requires the availability of accurate estimates of geometrical critical parameters, such as percolation thresholds and correlation length exponents, here we use Monte Carlo simulations to obtain these data for LLs when a site (or bond) is present with probability p. We have found that the thresholds satisfy a mean-field (Bethe lattice) trend, namely that the critical concentration, p_{c}, increases as the average coordination number decreases; our estimates for the correlation length exponent are in line with the expectation that there is no change in the universality class.

Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits

It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study (2+1)D random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to (2+1)D percolation, a possibly stable critical phase, topologically ordered, trivial, and volume law phases, and lines of critical points separating them. With Pauli-Y single qubit measurements instead, we find an anisotropic self-dual tricritical point, with dynamical exponent z≈1.46, exhibiting logarithmic violation of the area law and an anomalous exponent for the topological entanglement entropy, which thus appears distinct from any known percolation fixed point. The phase diagram also hosts a measurement-induced volume law entangled phase in the absence of unitary dynamics.

An upper bound for the bond percolation threshold of the cubic lattice by a growth process approach

We reduce the upper bound for the bond percolation threshold of the cubic lattice from 0.447 792 to 0.347 297. The bound is obtained by a growth process approach which views the open cluster of a bond percolation model as a dynamic process. A three-dimensional dynamic process on the cubic lattice is constructed and then projected onto a carefully chosen plane to obtain a two-dimensional dynamic process on a triangular lattice. We compare the bond percolation models on the cubic lattice and their projections, and demonstrate that the bond percolation threshold of the cubic lattice is no greater than that of the triangular lattice. Applying the approach to the body-centered cubic lattice yields an upper bound of 0.292 893 for its bond percolation threshold.

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δ_{c}. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δ_{c} is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)10.1016/j.advwatres.2020.103565] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ^{2}〉, which is easier to determine than δ_{c}. Here, we compare δ_{c} to the second moment of the pore size 〈δ^{2}〉, and indeed confirm that, for all porosities and all models considered, δ_{c}^{2} is to a good approximation proportional to 〈δ^{2}〉. However, unlike 〈δ^{2}〉, the permeability estimate based on δ_{c}^{2} does not predict the correct ranking of k for our models. Thus, we confirm 〈δ^{2}〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

Percolation of the two-dimensional XY model in the flow representation

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.

Power-law bounds for critical long-range percolation below the upper-critical dimension

We study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

History-dependent percolation in two dimensions

We study the history-dependent percolation in two dimensions, which evolves in generations from standard bond-percolation configurations through iteratively removing occupied bonds. Extensive simulations are performed for various generations on periodic square lattices up to side length L=4096. From finite-size scaling, we find that the model undergoes a continuous phase transition, which, for any finite number of generations, falls into the universality of standard two-dimensional (2D) percolation. At the limit of infinite generation, we determine the correlation-length exponent 1/ν=0.828(5) and the fractal dimension d_{f}=1.8644(7), which are not equal to 1/ν=3/4 and d_{f}=91/48 for 2D percolation. Hence, the transition in the infinite-generation limit falls outside the standard percolation universality and differs from the discontinuous transition of history-dependent percolation on random networks. Further, a crossover phenomenon is observed between the two universalities in infinite and finite generations.

Applications of a neural network to detect the percolating transitions in a system with variable radius of defects

We systematically study the percolation phase transition at the change of concentration of the chaotic defects (pores) in an extended system where the disordered defects additionally have a variable random radius, using the methods of a neural network (NN). Two important parameters appear in such a material: the average value and the variance of the random pore radius, which leads to significant change in the properties of the phase transition compared with conventional percolation. To train a network, we use the spatial structure of a disordered environment (feature class), and the output (label class) indicates the state of the percolation transition. We found high accuracy of the transition prediction (except the narrow threshold area) by the trained network already in the two-dimensional case. We have also employed such a technique for the extended three-dimensional (3D) percolation system. Our simulations showed the high accuracy of prediction in the percolation transition in 3D case too. The considered approach opens up interesting perspectives for using NN to identify the phase transitions in real percolating nanomaterials with a complex cluster structure.

Bond percolation on simple cubic lattices with extended neighborhoods

We study bond percolation on the simple cubic lattice with various combinations of first, second, third, and fourth nearest neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111. However, for large z, the threshold must approach the Bethe lattice result p_{c}=1/(z-1). Fitting our data and data for additional nearest neighbors, we find p_{c}(z-1)=1+1.224z^{-1/2}.

Percolation of Fortuin-Kasteleyn clusters for the random-bond Ising model

We apply generalizations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional ±1 random-bond Ising model. The behavior of the model is determined by the temperature T and the concentration p of negative (antiferromagnetic) bonds. The ground state is ferromagnetic for 0≤p<p_{c}, and a spin glass for p_{c}<p≤0.5 where p_{c}≃0.222. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as a function of temperature for large system sizes up to N=200^{3} spins. Except for p=0 the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for p=1/2 but here we provide evidence for a difference in transition temperatures even for p arbitrarily small. Furthermore, for all values of p>0, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for p=0 where the clusters are strictly tied to Ising correlations so the percolation transition is in the Ising universality class.

Theory of relaxor-ferroelectricity

Relaxor-ferroelectrics are fascinating and useful materials, but the mechanism of relaxor-ferroelectricity has been puzzling the scientific community for more than 65 years. Here, a theory of relaxor-ferroelectricity is presented based on 3-dimensional-extended-random-site-Ising-model along with Glauber-dynamics of pseudospins. We propose a new mean-field of pseudospin-strings to solve this kinetic model. The theoretical results show that, with decreasing pseudospin concentration, there are evolutions from normal-ferroelectrics to relaxor-ferroelectrics to paraelectrics, especially indicating by the crossovers from, (a) the sharp to diffuse change at the phase-transition temperature to disappearance in the whole temperature range of order-parameter, and (b) the power-law to Vogel-Fulcher-law to Arrhenius-relation of the average relaxation time. Particularly, the calculated local-order-parameter of the relaxor-ferroelectrics gives the polar-nano-regions appearing far above the diffuse-phase-transition and shows the quasi-fractal characteristic near and below the transition temperature. We also provide a new mechanism of Burns-transformation which stems from not only the polar-nano-regions but also the correlation-function between pseudospins, and put forward a definition of the canonical relaxor-ferroelectrics. The theory accounts for the main facts of relaxor-ferroelectricity, and in addition gives a good quantitative agreement with the experimental results of the order-parameter, specific-heat, high-frequency permittivity, and Burns-transformation of lead magnesium niobate, the canonical relaxor-ferroelectric.

The critical radius in sampling-based motion planning

We develop a new analysis of sampling-based motion planning in Euclidean space with uniform random sampling, which significantly improves upon the celebrated result of Karaman and Frazzoli and subsequent work. In particular, we prove the existence of a critical connection radius proportional to [Formula: see text] for n samples and d dimensions: below this value the planner is guaranteed to fail (similarly shown by Karaman and Frazzoli). More importantly, for larger radius values the planner is asymptotically (near-)optimal. Furthermore, our analysis yields an explicit lower bound of [Formula: see text] on the probability of success. A practical implication of our work is that asymptotic (near-)optimality is achieved when each sample is connected to only [Formula: see text] neighbors. This is in stark contrast to previous work that requires [Formula: see text] connections, which are induced by a radius of order [Formula: see text]. Our analysis applies to the probabilistic roadmap method (PRM), as well as a variety of “PRM-based” planners, including RRG, FMT*, and BTT. Continuum percolation plays an important role in our proofs. Lastly, we develop similar theory for all the aforementioned planners when constructed with deterministic samples, which are then sparsified in a randomized fashion. We believe that this new model, and its analysis, is interesting in its own right.

Random sequential adsorption of lattice animals on a three-dimensional cubic lattice

The properties of the random sequential adsorption of objects of various shapes on simple three-dimensional (3D) cubic lattice are studied numerically by means of Monte Carlo simulations. Depositing objects are "lattice animals," made of a certain number of nearest-neighbor sites on a lattice. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density θ_{J} and on the temporal evolution of the coverage fraction θ(t). We analyzed all lattice animals of size n=1, 2, 3, 4, and 5. A significant number of objects of size n⩾6 were also used to confirm our findings. Approach of the coverage θ(t) to the jamming limit θ_{J} is found to be exponential, θ_{J}-θ(t)∼exp(-t/σ), for all lattice animals. It was shown that the relaxation time σ increases with the number of different orientations m that lattice animals can take when placed on a cubic lattice. Orientations of the lattice animal deposited in two randomly chosen places on the lattice are different if one of them cannot be translated into the other. Our simulations performed for large collections of 3D objects confirmed that σ≅m∈{1,3,4,6,8,12,24}. The presented results suggest that there is no correlation between the number of possible orientations m of the object and the corresponding values of the jamming density θ_{J}. It was found that for sufficiently large objects, changing of the shape has considerably more influence on the jamming density than increasing of the object size.

Statistics of correlated percolation in a bacterial community

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.

Percolation clusters of organics in interstellar ice grains as the incubators of life

Biomolecules can be synthesized in interstellar ice grains subject to UV radiation and cosmic rays. I show that on time scales of ≳10⁶ years, these processes lead to the formation of large percolation clusters of organic molecules. Some of these clusters would have ended up on proto-planets where large, loosely bound aggregates of clusters (superclusters) would have formed. The interior regions of such superclusters provided for chemical micro-environments that are filtered versions of the outside environment. I argue that models for abiogenesis are more likely to work when considered inside such micro-environments. As the supercluster breaks up, biochemical systems in such micro-environments gradually become subject to a less filtered environment, allowing them to get adapted to the more complex outside environment. A particular system originating from a particular location on some supercluster would have been the first to get adapted to the raw outside environment and survive there, thereby becoming the first microbe. A collision of a microbe-containing proto-planet with the Moon could have led to fragments veering off back into space, microbes in small fragments would have been able to survive a subsequent impact with the Earth.

Critical p=1/2 in percolation on semi-infinite strips

We study site percolation on lattices confined to a semi-infinite strip. For triangular and square lattices we find that the probability that a cluster touches the three sides of such a system at the percolation threshold has a continuous limit of 1/2 and argue that this limit is universal for planar systems. This value is also expected to hold for finite systems for any self-matching lattice. We attribute this result to the asymptotic symmetry of the separation lines between alternating spanning clusters of occupied and unoccupied sites formed on the original and matching lattice, respectively.

Observation of nonscalar and logarithmic correlations in two- and three-dimensional percolation

In bulk percolation, we exhibit operators that insert N clusters with any given symmetry under the symmetric group S_{N}. At the critical threshold, this leads to predictions that certain combinations of two-point correlation functions depend logarithmically on distance, without the usual power law. The behavior under rotations of certain amplitudes of correlators is also determined exactly. All these results hold in any dimension, 2≤d≤6. Moreover, in d=2 the critical exponents and universal logarithmic prefactors are obtained exactly. We test these predictions against extensive simulations of critical bond percolation in d=2 and 3, for all correlators up to N=4 (d=2) and N=3 (d=3), finding excellent agreement. In d=3 we further obtain precise numerical estimates for critical exponents and logarithmic prefactors.

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as K_{c}(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C_{1} follows a single-variable function as P(C_{1},L)dC_{1}=P[over ̃](x)dx with x≡C_{1}/L^{d_{F}} (L is the linear size), where the fractal dimension d_{F} is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution P[over ̃](x) in three dimensions, and attributed to the different approaching behaviors for K→K_{c}+0^{±}. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as d_{min}(3D)=1.25936(12) and d_{min}(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as d_{B}(3D)=2.1673(15) and d_{B}(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents d_{min} and d_{B} either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.

Percolation in a distorted square lattice

This paper presents a Monte Carlo study of percolation in a distorted square lattice, in which the adjacent sites are not equidistant. Starting with an undistorted lattice, the position of the lattice sites are shifted through a tunable parameter α to create a distorted empty lattice. In this model, two occupied neighboring sites are considered to be connected to each other in order to belong to the same cluster, if the distance between them is less than or equal to a certain value, called connection threshold d. While spanning becomes difficult in distorted lattices as is manifested by the increment of the percolation threshold p_{c} with α, an increased connection threshold d makes it easier for the system to percolate. The scaling behavior of the order parameter studied through relevant critical exponents, and the fractal dimension d_{f} of the percolating cluster at p_{c} suggest that this new type of percolation may belong to the same universality class as ordinary percolation. This model can be very useful in various realistic applications since it is almost impossible to find a natural system that is perfectly ordered.

Percolation through Voids around Randomly Oriented Polyhedra and Axially Symmetric Grains

Porous materials made up of impermeable grains constrain fluid flow to voids around the impenetrable inclusions. A percolation transition marks the boundary between densities of grains permitting bulk transport and concentrations blocking traversal on macroscopic scales. With dynamical infiltration of void spaces using virtual tracer particles, we treat inclusion geometries exactly. We calculate the critical number density per volume ρ_{c} for a variety of axially symmetric shapes and faceted solids with the former including cylinders, ellipsoids, cones, and tablet shaped grains from highly oblate (platelike) to highly prolate (needlelike) extremes. For the latter, results suggest a common asymptotic value identical to the counterpart for aligned cylindrical grains. We find percolation thresholds for each of the five platonic solids (i.e., tetrahedra, cubes, octahedra, dodecahedra, and icosahedra) as well as truncated icosahedra. For each polyhedron type, we consider aligned and randomly oriented grains, finding distinct percolation thresholds for the former versus the latter only for cubes. The anomalous diffusion exponents we find differ from those of the universality class for discrete models on 3D lattices.

Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems

With the advances in artificial particle synthesis, it is possible to create particles with unique shapes. Particle shape becomes a feasible parameter for tuning the percolation behavior. How to accurately predict the percolation threshold by particle characteristics for arbitrary particles has aroused great interest. Towards this end, a versatile family of cuboidlike particles and a numerical contact detection algorithm for these particles are presented here. Then, combining with percolation theory, the continuum percolation of randomly distributed overlapping cuboidlike particles is studied. The global percolation threshold ϕ_{c} of overlapping particles with broad ranges of the shape parameter m in [1.0,+∞) and aspect ratio a/b in [0.1, 10.0] is computed via a finite-size scaling technique. Using the generalized excluded-volume approximation, an analytical formula is proposed to quantify the dependence of ϕ_{c} on the parameters m and a/b, and its reliability is verified. The results reveal that the percolation threshold ϕ_{c} of overlapping cuboidlike particles is heavily dependent on the shapes of particles, and much more sensitive to a/b than m. As the cuboidlike particles become spherical (i.e., m=1.0 and a/b=1.0), the maximum threshold ϕ_{c,max} can be obtained.

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

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.

Continuous Easy-Plane Deconfined Phase Transition on the Kagome Lattice

We use large scale quantum Monte Carlo simulations to study an extended Hubbard model of hard core bosons on the kagome lattice. In the limit of strong nearest-neighbor interactions at 1/3 filling, the interplay between frustration and quantum fluctuations leads to a valence bond solid ground state. The system undergoes a quantum phase transition to a superfluid phase as the interaction strength is decreased. It is still under debate whether the transition is weakly first order or represents an unconventional continuous phase transition. We present a theory in terms of an easy plane noncompact CP^{1} gauge theory describing the phase transition at 1/3 filling. Utilizing large scale quantum Monte Carlo simulations with parallel tempering in the canonical ensemble up to 15552 spins, we provide evidence that the phase transition is continuous at exactly 1/3 filling. A careful finite size scaling analysis reveals an unconventional scaling behavior hinting at deconfined quantum criticality.

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.

Key issues review: evolution on rugged adaptive landscapes

Adaptive landscapes represent a mapping between genotype and fitness. Rugged adaptive landscapes contain two or more adaptive peaks: allele combinations with higher fitness than any of their neighbors in the genetic space. How do populations evolve on such rugged landscapes? Evolutionary biologists have struggled with this question since it was first introduced in the 1930s by Sewall Wright. Discoveries in the fields of genetics and biochemistry inspired various mathematical models of adaptive landscapes. The development of landscape models led to numerous theoretical studies analyzing evolution on rugged landscapes under different biological conditions. The large body of theoretical work suggests that adaptive landscapes are major determinants of the progress and outcome of evolutionary processes. Recent technological advances in molecular biology and microbiology allow experimenters to measure adaptive values of large sets of allele combinations and construct empirical adaptive landscapes for the first time. Such empirical landscapes have already been generated in bacteria, yeast, viruses, and fungi, and are contributing to new insights about evolution on adaptive landscapes. In this Key Issues Review we will: (i) introduce the concept of adaptive landscapes; (ii) review the major theoretical studies of evolution on rugged landscapes; (iii) review some of the recently obtained empirical adaptive landscapes; (iv) discuss recent mathematical and statistical analyses motivated by empirical adaptive landscapes, as well as provide the reader with instructions and source code to implement simulations of evolution on adaptive landscapes; and (v) discuss possible future directions for this exciting field.

Universal features of cluster numbers in percolation

The number of clusters per site n(p) in percolation at the critical point p=p_{c} is not itself a universal quantity; it depends upon the lattice and percolation type (site or bond). However, many of its properties, including finite-size corrections, scaling behavior with p, and amplitude ratios, show various degrees of universal behavior. Some of these are universal in the sense that the behavior depends upon the shape of the system, but not lattice type. Here, we elucidate the various levels of universality for elements of n(p) both theoretically and by carrying out extensive studies on several two- and three-dimensional systems, by high-order series analysis, Monte Carlo simulation, and exact enumeration. We find many results, including precise values for n(p_{c}) for several systems, a clear demonstration of the singularity in n^{''}(p), and metric scale factors. We make use of the matching polynomial of Sykes and Essam to find exact relations between properties for lattices and matching lattices. We propose a criterion for an absolute metric factor b based upon the singular behavior of the scaling function, rather than a relative definition of the metric that has previously been used.

Percolation via Combined Electrostatic and Chemical Doping in Complex Oxide Films

Stimulated by experimental advances in electrolyte gating methods, we investigate theoretically percolation in thin films of inhomogeneous complex oxides, such as La_{1-x}Sr_{x}CoO_{3} (LSCO), induced by a combination of bulk chemical and surface electrostatic doping. Using numerical and analytical methods, we identify two mechanisms that describe how bulk dopants reduce the amount of electrostatic surface charge required to reach percolation: (i) bulk-assisted surface percolation and (ii) surface-assisted bulk percolation. We show that the critical surface charge strongly depends on the film thickness when the film is close to the chemical percolation threshold. In particular, thin films can be driven across the percolation transition by modest surface charge densities. If percolation is associated with the onset of ferromagnetism, as in LSCO, we further demonstrate that the presence of critical magnetic clusters extending from the film surface into the bulk results in considerable enhancement of the saturation magnetization, with pronounced experimental consequences. These results should significantly guide experimental work seeking to verify gate-induced percolation transitions in such materials.

Interstellar Travel and Galactic Colonization: Insights from Percolation Theory and the Yule Process

In this paper, percolation theory is employed to place tentative bounds on the probability p of interstellar travel and the emergence of a civilization (or panspermia) that colonizes the entire Galaxy. The ensuing ramifications with regard to the Fermi paradox are also explored. In particular, it is suggested that the correlation function of inhabited exoplanets can be used to observationally constrain p in the near future. It is shown, by using a mathematical evolution model known as the Yule process, that the probability distribution for civilizations with a given number of colonized worlds is likely to exhibit a power-law tail. Some of the dynamical aspects of this issue, including the question of timescales and generalizing percolation theory, were also studied. The limitations of these models, and other avenues for future inquiry, are also outlined.

KEY WORDS

Complex life-Extraterrestrial life-Panspermia-Life detection-SETI. Astrobiology 16, 418-426.

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.

Efficient Cluster Algorithm for Spin Glasses in Any Space Dimension

Spin systems with frustration and disorder are notoriously difficult to study, both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here, we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine.

Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

In this paper, random-site percolation thresholds for a simple cubic (SC) lattice with site neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation (Bastas et al., arXiv:1411.5834) is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are p(C)(4NN)=0.31160(12),p(C)(4NN+NN)=0.15040(12),p(C)(4NN+2NN)=0.15950(12),p(C)(4NN+3NN)=0.20490(12),p(C)(4NN+2NN+NN)=0.11440(12),p(C)(4NN+3NN+NN)=0.11920(12),p(C)(4NN+3NN+2NN)=0.11330(12), and p(C)(4NN+3NN+2NN+NN)=0.10000(12), where 3NN, 2NN, and NN stand for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with a lattice constant that is twice as large, the percolation threshold p(C)(4NN) is exactly equal to p(C)(NN). The simplified method of Bastas et al. allows for uncertainty of the percolation threshold value p(C) to be reached, similar to that obtained with the classical method but ten times faster.

Leaf-excluded percolation in two and three dimensions

We introduce the leaf-excluded percolation model, which corresponds to independent bond percolation conditioned on the absence of leaves (vertices of degree one). We study the leaf-excluded model on the square and simple-cubic lattices via Monte Carlo simulation, using a worm-like algorithm. By studying wrapping probabilities, we precisely estimate the critical thresholds to be 0.3552475(8) (square) and 0.185022(3) (simple-cubic). Our estimates for the thermal and magnetic exponents are consistent with those for percolation, implying that the phase transition of the leaf-excluded model belongs to the standard percolation universality class.

Short-range correlations in percolation at criticality

We derive the critical nearest-neighbor connectivity gn as 3/4, 3(7-9pc(tri))/4(5-4pc(tri)), and 3(2+7pc(tri))/4(5-pc(tri)) for bond percolation on the square, honeycomb, and triangular lattice, respectively, where pc(tri)=2sin(π/18) is the percolation threshold for the triangular lattice, and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as gnn=0.6875000(2), which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006], implying the exact value gnn=11/16. We also determine the connectivity on a free surface as gn(surf)=0.6250001(13) and conjecture that this value is exactly equal to 5/8. In addition, we find that at criticality, the connectivities depend on the linear finite size L as ∼L(yt-d), and the associated specific-heat-like quantities Cn and Cnn scale as ∼L(2yt-d)ln(L/L0), where d is the lattice dimensionality, yt=1/ν the thermal renormalization exponent, and L0 a nonuniversal constant. We provide an explanation of this logarithmic factor within the theoretical framework reported recently by Vasseur et al. [J. Stat. Mech. (2012) L07001].

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.

Geometric structure of percolation clusters

We investigate the geometric properties of percolation clusters by studying square-lattice bond percolation on the torus. We show that the density of bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch iff at least one of the two clusters produced by its deletion is a tree. Starting from a percolation configuration and deleting the branches results in a leaf-free configuration, whereas, deleting all bridges produces a bridge-free configuration. Although branches account for ≈43% of all occupied bonds, we find that the fractal dimensions of the cluster size and hull length of leaf-free configurations are consistent with those for standard percolation configurations. By contrast, we find that the fractal dimensions of the cluster size and hull length of bridge-free configurations are given by the backbone and external perimeter dimensions, respectively. We estimate the backbone fractal dimension to be 1.643 36(10).