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Percolation in an ultrametric space

   
@article{EJP1789,
	author = {Donald Dawson and Luis Gorostiza},
	title = {Percolation in an ultrametric space},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {18},
	year = {2013},
	keywords = {Percolation; hierarchical graph; ultrametric; renormalization},
	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 ck/Nk(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 percolation occurs, 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-Rényi random graphs play  a key role. We find sufficient conditions on ck  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 ck. Those questions  suggest the need to study ultrametric random graphs. 
 
 
},
	pages = {no. 12, 1-26},
	issn = {1083-6489},
	doi = {10.1214/EJP.v18-1789},    
        url = {http://ejp.ejpecp.org/article/view/1789}}