Abstract
We study percolation in the hierarchical lattice of order N where the probability of connection between two points separated by distance k is of the form c k / N k ( 1+δ) , δ>-1. Since the distance is an ultrametric , there are significant differences with percolation in the Euclidean lattice. We consider three regimes: δ 1 , where it does not occur and δ=1 which is the critical case corresponding to the phase transition. In the critical case we use an approach in the spirit of the renormalization group method of statistical physics, and connectivity results of Erdős-Renyi random graphs play a key role. We find sufficient conditions on c k such that percolation occurs, or that it does not occur. An intermediate situation called pre-percolation, which is necessary for percolation, is also considered. In the cases of percolation we prove uniqueness of the constructed percolation clusters. In a previous paper we studied percolation in the N→∞ limit (mean field percolation), which provided a simplification that allowed finding a necessary and sufficient condition for percolation. For fixed N there are open questions, in particular regarding the behaviour at the critical values of parameters in the definition of c k . Those questions suggest the need to study ultrametric random graphs .
Highlights
Percolation theory in a lattice began with the work of Broadbent and Hammersley in 1957
In [16] we studied asymptotic percolation in ΩN as N → ∞ with connection probabilities of the form ck/N 2k−1 between two points separated by distance k, and we obtained a necessary and sufficient condition for percolation. (See Subsection 3.1 for the definition of asymptotic percolation)
In the present paper we study percolation in ΩN for fixed N with connection probabilities of the form ck/N k(1+δ), δ > −1, between two points separated by distance k
Summary
Percolation theory in a lattice (e.g., the Euclidean lattice Zd) began with the work of Broadbent and Hammersley in 1957. In the present paper we study percolation in ΩN for fixed N with connection probabilities of the form ck/N k(1+δ), δ > −1, between two points separated by distance k In this case percolation means that there is a positive probability that a given point of ΩN belongs to an infinite connected component. While we were working on this paper we learned about the manuscript of Koval et al [26] (for which we thank them), where they study percolation in ΩN for fixed N , with connection probabilities of the form 1 − exp{−α/βk}, α ≥ 0, β > 0, between two points separated by distance k Some of their results for β > 1 and ours may be compared by setting β = N 1+δ (see Remark 3.2). In an appendix we give a result on connectivity of random graphs derived from [19], which is a key ingredient for the proof of percolation in the critical case
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