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

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.