Cluster tomography in percolation

Abstract

In cluster tomography, we propose measuring the number of clusters $N$ intersected by a line segment of length $\ell$ across a finite sample. As expected, the leading order of $N(\ell )$ scales as $a\ell$, where $a$ depends on microscopic details of the system. However, at criticality, there is often an additional nonlinearity of the form $b\ln (\ell )$, originating from the endpoints of the line segment. By performing large scale Monte Carlo simulations of both 2$d$ and 3$d$ percolation, we find that $b$ is universal and depends only on the angles encountered at the endpoints of the line segment intersecting the sample. Our findings are further supported by analytic arguments in 2$d$, building on results in conformal field theory. Being broadly applicable, cluster tomography can be an efficient tool to detect phase transitions and to characterize the corresponding universality class in classical or quantum systems with a relevant cluster structure.

11 pages, 7 figures

References in corpus (5)

• Novel type of phase transition in a system of self-driven particles
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Entanglement Entropy in the Two-Dimensional Random Transverse Field Ising Model
• Corner contribution to percolation cluster numbers
• Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit