Cluster tomography in percolation

TL;DR

This paper introduces cluster tomography as a method to analyze percolation systems, revealing universal endpoint effects at criticality through simulations and conformal field theory insights.

Contribution

It demonstrates that the endpoint-induced logarithmic correction in cluster intersection counts is universal and depends only on intersection angles, supported by simulations and analytic arguments.

Findings

The coefficient of the logarithmic term is universal.

Endpoint angles determine the correction magnitude.

Method applicable to classical and quantum systems.

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.