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

Abstract

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.

Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs

“…Third type is an abrupt singularity, where the order parameter changes discontinuously [22]. Similar discontinuous transition is observed in the numerical simulation of the hyperbolic lattice [23], which has a dual relation to the so-called infinite-order transition known in the Cayley trees [24][25][26][27]. All of these types of singularity are observed in the single system, where the two types of the graph-growth rules are randomly mixed, by tuning the mixing ratio [28].…”

“…Particularly, lim n→∞ 1 − ψ n = 0 for p > p c2 . Equation (27) gives the relation between ψ n and m. Thus, if we know the relation between ψ n and the RG solution g n , we can understand the relation between the RG behavior and the singularity of the order parameter.…”