Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

Abstract

Abstract. We examine the interplay between anisotropy and percolation, i. e., the spontaneous formation of a system spanning cluster in an anisotropic model. We simulate an extension of a benchmark model of continuum percolation, the Boolean model, which is formed by overlapping grains. Here we introduce an orientation bias of the grains that controls the degree of anisotropy of the generated patterns. We analyze in the Euclidean plane the percolation thresholds above which percolating clusters in x-and in y-direction emerge. Only in finite systems, distinct differences between effective percolation thresholds for different directions appear. If extrapolated to infinite system sizes, these differences vanish independent of the details of the model. In the infinite system, the uniqueness of the percolating cluster guarantees a unique percolation threshold. While percolation is isotropic even for anisotropic processes, the value of the percolation threshold depends on the model parameters, which we explore by simulating a score of models with varying degree of anisotropy. To which precision can we predict the percolation threshold without simulations? We discuss analytic formulas for approximations (based on the excluded area or the Euler characteristic) and compare them to our simulation results. Empirical parameters from similar systems allow for accurate predictions of the percolation thresholds (with deviations of < 5% in our examples), but even without any empirical parameters, the explicit approximations from integral geometry provide, at least for the systems studied here, lower bounds that capture well the qualitative dependence of the percolation threshold on the system parameters (with deviations of 5%-30%). As an outlook, we suggest further candidates for explicit and geometric approximations based on second moments of the so-called Minkowski functionals.

Influence of disorder on transfer characteristics of organic electrochemical transistors

“…36,39,40 The critical bond connectivity, g, extracted by the proposed model for p(g2T-TT) is about one order of magnitude smaller than expected for close-packed spheres, and for PEDOT:PSS, it is about two orders of magnitude smaller than expected. 29 However, g decreases strongly as the aspect ratio of conducting structures increases, 41 and conduction in polymers depends on high aspect ratio structures due to carrier delocalization along polymer backbones 42 and anisotropic phase segregation. 43 Therefore, g is expected to be much smaller in polymers than in systems composed of close-packed spheres.…”

Geometric functionals of fractal percolation

“…In view of the aforementioned observations in [12,[19][20][21], that the zero of the expected Euler characteristic per site is close to the percolation thresholds in many percolation models, let us now discuss the connections between the limit functionals for F introduced above and the connectivity properties in fractal percolation. p 0 is a lower bound for p c .…”

“…Morphometric methods to estimate thresholds in percolation models have been proposed in [20] and intensively studied in the physics literature [12,19,21,22]; see also the recent study in homological percolation [1] using topological data analysis. These methods are based on additive functionals from integral geometry, in particular the Euler characteristic, and rely on the observation that in many percolation models the expected Euler characteristic per site (as a function of the model parameter p)-which can easily be computed analytically in many models-has a zero close to the percolation threshold of the model.…”

“…on the thresholds that capture their dependence on system parameters such as the degree of anisotropy [12]. More precisely, across many models, the zero of the Euler characteristic has been observed to be a lower bound for the percolation threshold p c if p c < 1 2 , and to be an upper bound for p c if p c > 1 2 .…”

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“…Prior work suggests that the zeros of the Euler characteristic can be used to estimate the percolation threshold of a two-phase system [51][52][53][54]. We see from Eq.…”