Exactly solvable percolation problems

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

Study on Seepage Characteristics of Grouting Slurry for Water-Absorbing Mudstone with Rough Fissure

Based on the computed tomography scanning, which abbreviation is CT scanning, and fractal theory, geometric parameters of mudstone fissures are obtained. The physical model of a single fissured channel is obtained in combination with Barton standard curves and 3D printing technology, and similar materials of mudstone are developed based on the water absorption of natural mudstone to prepare single fissured water-absorbing grouting test blocks with different roughness levels for the grouting simulation testing. By analyzing the viscosity change characteristics of grouting slurry before and after grouting, the seepage characteristics of the grouting slurry in the rough fissures of the water-absorbing mudstone are revealed. The results show that when the roughness is small, the grouting slurry will have an obvious water loss effect after passing through mudstone fissures. However, with the flow of the slurry, the water loss effect of the subsequent grouting slurry will be weakened. For fissures with high roughness, the water absorption properties of the rough surfaces and the walls of the mudstone fissures work together, leading to the sedimentation and blockage of the fissure channels, thereby hindering the flow of slurry.

Strong connectivity in real directed networks

In many real, directed networks, the strongly connected component of nodes which are mutually reachable is very small. This does not fit with current theory, based on random graphs, according to which strong connectivity depends on mean degree and degree-degree correlations. And it has important implications for other properties of real networks and the dynamical behavior of many complex systems. We find that strong connectivity depends crucially on the extent to which the network has an overall direction or hierarchical ordering-a property measured by trophic coherence. Using percolation theory, we find the critical point separating weakly and strongly connected regimes and confirm our results on many real-world networks, including ecological, neural, trade, and social networks. We show that the connectivity structure can be disrupted with minimal effort by a targeted attack on edges which run counter to the overall direction. This means that many dynamical processes on networks can depend significantly on a small fraction of edges.

Random site percolation on honeycomb lattices with complex neighborhoods

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3-24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number z of sites in the neighborhoods, the site percolation thresholds _p follow the dependency _p ∝ 1 / z, as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of _p corresponding to the same number z of sites in the neighborhoods). An example of a single-value index ζ = ∑ i _{z r}-where _z and _r are the number of sites and radius of the ith coordination zone, respectively-characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence _p ∝ 1 / ζ. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.