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.
Highlights
1.1 Background and motivationsA lot of progress has been made in the past decade toward the understanding of statistical physics models in dimension 2
We study in more detail independent percolation on the uniform infinite planar quadrangulation (UIPQ) and the uniform infinite planar map (UIPM)
For our purpose, when dealing with quadrangulations, it turns out to be more convenient to work with faces rather than with edges, which leads us to introduce the new distance d on M defined by d (m, m ) = (1 + sup {r : Br (m) = Br (m )})−1 for all rooted maps m, m, where, for r 1, we denote by Br (m) the planar map obtained as the union of all faces of m that have at least one vertex at distance strictly smaller than r from the root
Summary
A lot of progress has been made in the past decade toward the understanding of statistical physics models in dimension 2. This led to the derivation of the so-called “arm exponents”, that describe the probability of observing disjoint long-range paths: for instance, at criticality, the probability for a given vertex to be connected to distance n follows a power law: it decays like n−α1+o(1) as n → ∞, with α1 =. Combining this new understanding with Kesten’s scaling relations [19], one can describe the. We study in more detail independent percolation on the UIPQ and the UIPM
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