Short-range correlations in percolation at criticality

Sweeny dynamics for the random-cluster model with small Q

The Sweeny algorithm for the Q-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As Q decreases, the so-called critical speeding-up for nonlocal quantities becomes more and more pronounced. However, for some quantity of a specific local pattern, e.g., the number of half faces on the square lattice, we observe that, as Q→0, the integrated autocorrelation time τ diverges as Q^{-ζ}, with ζ≃1/2, leading to the nonergodicity of the Sweeny method for Q→0. Such Q-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight v=sqrt[Q], can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order O(1).

Aerosol-Printed MoS2 Ink as a High Sensitivity Humidity Sensor

Molybdenum disulfide (MoS2) is attractive for use in next-generation nanoelectronic devices and exhibits great potential for humidity sensing applications. Herein, MoS2 ink was successfully prepared via a simple exfoliation method by sonication. The structural and surface morphology of a deposited ink film was analyzed by scanning electron microscopy (SEM), Raman spectroscopy, and atomic force microscopy (AFM). The aerosol-printed MoS2 ink sensor has high sensitivity, with a conductivity increase by 6 orders of magnitude upon relative humidity increase from 10 to 95% at room temperature. The sensor also has fast response/recovery times and excellent repeatability. Possible mechanisms for the water-induced conductivity increase are discussed. An analytical model that encompasses two ionic conduction regimes, with a percolation transition to an insulating state below a low humidity threshold, describes the sensor response successfully. In conclusion, our work provides a low-cost and straightforward strategy for fabricating a high-performance humidity sensor and fundamental insights into the sensing mechanism.

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.

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.

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.

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.

Recursive percolation

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