Existence of phase transition of percolation on Sierpiński carpet lattices

Abstract

We study Bernoulli bond percolation on Sierpiński carpet lattices, which is a class of graphs corresponding to generalized Sierpiński carpets. In this paper we give a sufficient condition for the existence of a phase transition on the lattices. The proof is suitable for graphs which have self-similarity. We also discuss the relation between the existence of a phase transition and the isoperimetric dimension.

Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result

“…However, the approximate calculations [2,4] and the numerical simulations [5] suggest that an Ising spin system on the Sierpinski carpet is spontaneously magnetized at finite temperature. Recently, this property has been proved in [12], where it is shown that it follows from the behaviour of percolation on the Sierpinski carpet. Here we give an alternative proof inspired by the classical Peierls argument [13] for the Ising model on the two-dimensional lattice and, in particular, by the version given in [14].…”

Remarks on Central Limit Theorems for the Number of Percolation Clusters

“…Shinoda [7] obtained a sufficient condition for T to satisfy p c (G T ) < 1. However, since G T is in general not periodic, it is difficult to know further properties, e.g.…”

“…While K T is an infinitely-ramified fractal, it is known that these crossing probabilities tend to zero as n → ∞ ( [4,7]). Thus we can obtain the CLT for p ∈ (0, 1).…”

Fractal hard drives for quantum information

“…To demonstrate this, Brell shows that the partition function of the new code corresponds to that of a simpler and well-studied model: the Ising model on a Sierpiński carpet. This is well known to have a magnetically ordered phase below a finite temperature phase transition [8][9][10]. For the Brell's code, this corresponds to a phase in which the qubit is protected from the noise caused by temperature.…”