We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive re...

We investigate the critical properties of Ising models on a regularized Apollonian network (RAN), here defined as a kind of Apollonian network in which the connectivity asymmetry associated with its corners is removed. Different choices for the coupling constants between nearest neighbors are considered and two different order parameters are used to detect the critical behavior. While ordinary ...

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z) fixing the Descartes quadratic form Q. For nonzero v ∈ Z4 satisfying Q(v) = 0, the orbits Pv = Av correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits Pv mod any integer d > 1. We show that this reduction...

Explicit determination of the mean first-passage time (MFPT) for trapping problem on complex media is a theoretical challenge. In this paper, we study random walks on the Apollonian network with a trap fixed at a given hub node (i.e. node with the highest degree), which are simultaneously scale-free and small-world. We obtain the precise analytic expression for the MFPT that is confirmed by dir...