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Abstract: We introduce and study an infinite random triangulation of the unit disk thatarises as the limit of several recursive models. This triangulation isgenerated by throwing chords uniformly at random in the unit disk and keepingonly those chords that do not intersect the previous ones. After throwinginfinitely many chords and taking the closure of the resulting set, one gets arandom compact subset of the unit disk whose complement is a countable union oftriangles. We show that this limiting random set has Hausdorff dimension$\beta^*+1$, where $\beta^*=\sqrt{17}-3-2$, and that it can be described asthe geodesic lamination coded by a random continuous function which isH\-{o}lder continuous with exponent $\beta^*-\varepsilon$, for every$\varepsilon>0$. We also discuss recursive constructions of triangulations ofthe $n$-gon that give rise to the same continuous limit when $n$ tends toinfinity.



Author: Nicolas Curien, Jean-Fran├žois Le Gall

Source: https://arxiv.org/







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