Large Boltzmann planar maps with a stable exponent alpha converge in the Gromov-Hausdorff-Prokhorov sense to an explicit random compact metric space S_alpha of Hausdorff dimension 2 alpha.
Looptree, Fennec, and Snake of ICRT
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abstract
We introduce a new theory of plane $\mathbb R$-tree, to define plane ICRT (inhomogeneous continuum random tree), and its looptree, fennec (a Gaussian free field on the looptree), and snake. We prove that a.s. the looptree is compact, and that a.s. the fennec and snake are continuous. We compute the looptree's fractal dimensions, and the fennec and snake's H\"older exponent. Alongside, we define a Gaussian free field on the ICRT, and prove a condition for its continuity. In a companion paper , we prove that the looptrees, fennecs, and snakes of trees with fixed degree sequence converge toward the looptrees, fennecs and snakes of ICRT.
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The scaling limit of planar maps with large faces
Large Boltzmann planar maps with a stable exponent alpha converge in the Gromov-Hausdorff-Prokhorov sense to an explicit random compact metric space S_alpha of Hausdorff dimension 2 alpha.