We construct and study the unique random tiling of the hyperbolic plane into ideal hyperbolic triangles (with the three corners located on the boundary) that is invariant (in law) with respect to M\"obius transformations, and possesses a natural spatial Markov property that can be roughly described as the conditional independence of the two parts of the triangulation on the two sides of the edge of one of its triangles.
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Wendelin Werner, Nicolas Curien, The Markovian hyperbolic triangulation. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1309–1341DOI 10.4171/JEMS/393