Cluster tomography in percolation

Abstract

In cluster tomography, we propose measuring the number of clusters N intersected by a line segment of length ℓ across a finite sample. As expected, the leading order of N(ℓ) scales as aℓ, where a depends on microscopic details of the system. However, at criticality, there is often an additional nonlinearity of the form bln(ℓ), originating from the endpoints of the line segment. By performing large scale Monte Carlo simulations of both two- and three-dimensional 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 two dimensions, building on results in conformal field theory. Being broadly applicable, cluster tomography can be an efficient tool for detecting phase transitions and characterizing the corresponding universality class in classical or quantum systems with a relevant cluster structure.