Percolation on uniform infinite planar maps

Abstract

We construct the uniform infinite planar map (UIPM), obtained as the n \to \infty local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is p_c^bond=1/3.

26 pages, 9 figures

References in corpus (4)

• Local limit of labeled trees and expected volume growth in a random quadrangulation
• The degree distribution in bipartite planar maps: applications to the Ising model
• A view from infinity of the uniform infinite planar quadrangulation
• Scaling of Percolation on Infinite Planar Maps, I

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• Geometry and percolation on half planar triangulations
• Simple peeling of planar maps with application to site percolation
• Random maps: proceeding of the Journ{é}es MAS 2014