Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree

TL;DR

This paper investigates the local cluster-size distribution in bond percolation on Cayley trees, revealing boundary effects and phase distinctions that differ from Euclidean lattices, and relates these to the Bethe lattice.

Contribution

It introduces a local perspective on phase characterization in Cayley trees, connecting boundary effects with the well-defined phases of Bethe lattices, and analyzes the distinct PDFs at the origin and boundary.

Findings

PDF at the origin is bimodal with finite-size scaling

PDF at the leaves follows a power law similar to critical Euclidean lattices

Boundary effects significantly influence local cluster-size distributions

Abstract

We study bond percolation of the Cayley tree (CT) by focusing on the probability distribution function (PDF) of a local variable, namely, the size of the cluster including a selected vertex. Because the CT does not have a dominant bulk region, which is free from the boundary effect, even in the large-size limit, the phase of the system on it is not well defined. We herein show that local observation is useful to define the phase of such a system in association with the well-defined phase of the system on the Bethe lattice, that is, an infinite regular tree without boundary. Above the percolation threshold, the PDFs of the vertex at the center of the CT (the origin) and of the vertices near the boundary of the CT (the leaves) have different forms, which are also dissimilar to the PDF observed in the ordinary percolating phase of a Euclidean lattice. The PDF for the origin of the CT is bimodal: a decaying exponential function and a system-size-dependent asymmetric peak, which obeys a finite-size-scaling law with a fractal exponent. These modes are respectively related to the PDFs of the finite and infinite clusters in the nonuniqueness phase of the Bethe lattice. On the other hand, the PDF for the leaf of the CT is a decaying power function. This is similar to the PDF observed at a critical point of a Euclidean lattice but is attributed to the nesting structure of the CT around the boundary.