Cluster tomography in percolation reveals universal information

Cluster formation is prevalent in complex systems, including magnetic domains, active matter, and cell migration. Characteristic properties of clusters often change when the system undergoes a phase transition. Consequently, phase transitions can, in principle, be detected through examining the cluster statistics. Here we demonstrate such an approach by investigating the statistics of cluster tomography, which measures the number of clusters N intersected by a line segment of length l through a finite sample. To leading order, N(l) scales as al, where a depends on microscopic details of the system. However, at criticality there is often an additional nonlinear term of the form bln(l), where b is universal. By investigating cluster tomography in 2d and 3d percolation using large-scale Monte Carlo simulations, we show that b depends only on the endpoint configurations of the line segment. In 2d the numerical results are further supported by analytic arguments from conformal field theory. More broadly, we demonstrate how cluster tomography can be an effective technique for identifying phase transitions in clustered systems and extracting universal information about the system at criticality.