# Anisotropy in finite continuum percolation: Threshold estimation by Minkowski functionals

## TL;DR

This paper investigates how anisotropy affects percolation thresholds in continuum models, using Minkowski functionals to estimate thresholds and analyzing finite versus infinite systems.

## Contribution

It introduces anisotropic extensions to the Boolean model and develops geometric approximation formulas for percolation thresholds based on Minkowski functionals.

## Key findings

- Percolation thresholds depend on anisotropy and can be accurately predicted with empirical parameters.
- Explicit geometric approximations provide reliable lower bounds for thresholds.
- Differences in thresholds for different directions vanish in infinite systems.

## 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.