# Circles of equal radii randomly placed on a plane: some rigorous results, asymptotic behavior, and application to transparent electrodes

## TL;DR

This paper derives exact and asymptotic distributions of arc lengths formed by randomly placed equal circles on a plane and applies these results to estimate the sheet resistance of transparent electrodes.

## Contribution

It provides rigorous results for arc length distributions and their asymptotic behavior, and demonstrates an application to transparent electrode design.

## Key findings

- Exact arc length distribution derived for intersecting circles.
- Asymptotic exponential distribution of arc angles in dense systems.
- Application to estimating sheet resistance of transparent electrodes.

## Abstract

We consider $N$ circles of equal radii, $r$, having their centers randomly placed within a square domain $\mathcal{D}$ of size $L \times L$ with periodic boundary conditions ($\mathcal{D} \in \mathbb{R}^2$). When two or more circles intersect each other, each circle is divided by the intersection points into several arcs. We found the exact length distribution of the arcs. In the limiting case of dense systems and large size of the domain $\mathcal{D}$ ($L \to \infty$ in such a way that the number of circle per unit area, $n=N/L^2$, is constant), the arc distribution approaches the probability density function (PDF) $f(\psi) = 4 n r^2\exp(-4 n r^2 \psi)$, where $\psi$ is the central angle subtended by the arc. This PDF is then used to estimate the sheet resistance of transparent electrodes based on conductive rings randomly placed onto a transparent insulating film.