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.

arXiv comment11 pages, 7 figures

References in corpus (18)

• Novel type of phase transition in a system of self-driven particles
• Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Critical Percolation in Finite Geometries
• Precise determination of the bond percolation thresholds and finite-size scaling corrections for the s.c., f.c.c., and b.c.c. lattices
• SLE for theoretical physicists
• Bond and Site Percolation in Three Dimensions
• Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation
• Entanglement Entropy in the Two-Dimensional Random Transverse Field Ising Model
• Universal logarithmic terms in the entanglement entropy of 2d, 3d and 4d random transverse-field Ising models
• Exact results at the 2-D percolation point
• Logarithmic corrections to the free energy from sharp corners with angle $2\pi$
• Corner contribution to percolation cluster numbers
• Entanglement susceptibilities and universal geometric entanglement entropy
• Corner contribution to cluster numbers in the Potts model
• Corner contribution to percolation cluster numbers in three dimensions
• Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit
• The expected number of critical percolation clusters intersecting a line segment