Long range percolation on the hierarchical lattice
2012 (English)In: Electronic Journal of Probability, ISSN 1083-6489, Vol. 17, 1-21 p.Article in journal (Refereed) Published
We study long-range percolation on the hierarchical lattice of order N, where any edge of length k is present with probability p(k) = 1 - exp (-beta(-k)alpha), independently of all other edges. For fixed beta, we show that alpha(c)(beta) ( the infimum of those alpha for which an infinite cluster exists a.s.) is non-trivial if and only if N < beta < N-2. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of alpha(c)(beta) as a function of beta. This means that the phase diagram of this model is well understood.
Place, publisher, year, edition, pages
2012. Vol. 17, 1-21 p.
long-range percolation, renormalisation, ergodicity
IdentifiersURN: urn:nbn:se:su:diva-80319DOI: 10.1214/EJP.v17-1977ISI: 000306787800001OAI: oai:DiVA.org:su-80319DiVA: diva2:552975