Abstract
A construction as a growth process for sampling of the uniform in- finite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT.
By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r 4 up to polylogarithmic factors, in accordance with heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r 2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon (also defined in [AnS]) converges in distribution to an asymmetric stable random variable of type 1/2.
By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability p c = 1/2 and that at p c percolation does not occur.
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Angel, O. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13, 935–974 (2003). https://doi.org/10.1007/s00039-003-0436-5
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DOI: https://doi.org/10.1007/s00039-003-0436-5