Scaling Limit of Critical Percolation Clusters on Hyperbolic Random Half‐Planar Triangulations and the Associated Random Walks

ABSTRACT We show that the Gromov‐Hausdorff‐Prohorov scaling limit of a critical percolation cluster on a random hyperbolic triangulation of the half‐plane is the Brownian continuum random tree. As a corollary, we obtain that a simple random walk on the critical cluster rescales to Brownian motion on the continuum random tree. As an intermediate result, we also obtain the Gromov‐Hausdorff‐Prohorov scaling limit of a general model of two‐type decorated Galton–Watson trees, under an exponential moment condition.