Percolation on uniform infinite planar maps

By Laurent M'enard, Pierre Nolin. 2013

52 citations recorded by the source corpus.

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 $p^{\textrm{bond}}_c=1/2$ and $p^{\textrm{site}}_c=2/3$ 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 $p^{\textrm{bond}}_c=1/3$.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c8e393affb761397acea4f13fa136039a45c598f

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Scaling limits for the peeling process on random maps (2014, methodology): “The peeling process has been used since to derive information about the metric properties of the UIPT [3], about percolation [3,4,26] and simple random walk [7] on the UIPT and its generalizations, and more recently about the conformal structure [15] of random planar maps.”
Planar stochastic hyperbolic triangulations (2016, methodology): “In the theory of random planar maps, the spatial Markov property is a key feature that has already been thoroughly used, generally under the form of the “peeling process” [4, 6, 12, 19, 21, 33].”
Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps (2014, background): “2 Uniform planar maps It has been shown in [22] that a uniform map chosen amongst the set of rooted maps with n edges converges locally in distribution.”