![]() ![]() The subcritical regime is also investigated. This establishes the first evidence of a phase transition for the topology of the boundary: in the dense phase, large faces are self-intersecting while in the dilute phase, they are self-avoiding. In this work, we complete the picture by proving that in the dilute phase $\alpha\in(3/2,2)$ (as well as in the generic critical regime), the scaling limit is a multiple of the unit circle. In the so-called dense phase $\alpha\in (1,3/2)$, it was established in that the scaling limit of the boundary is a stable looptree. We first deal with the non-generic critical regime, where the degree of a typical face falls within the domain of attraction of a stable law with parameter $\alpha \in (1,2)$. We study the scaling limits of the boundary of Boltzmann planar maps conditioned on having a large perimeter. We prove Benjamini-Schramm convergence of random $k$-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their $2$-connected components. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest $2$-connected component in random graphs from planar-like classes. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini-Schramm limit. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. ![]() Our main applications treat a selection of examples encompassed by this model. We study a model of random $\mathcal$-structures. Our proofs are based on recent results on the left tail behaviour of the martingale limit obtained by Fleischmann and Wachtel. This shows in particular that the conditioned trees converge to the regular (Formula presented.)-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy. More precisely, we show that if (Formula presented.) then with high probability, as (Formula presented.), (Formula presented.) takes exactly one or two values. ĪB - We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than (Formula presented.), agrees up to generation (Formula presented.) with a regular (Formula presented.)-ary tree, where (Formula presented.) is the essential minimum of the offspring distribution and the random variable (Formula presented.) is strongly concentrated near an explicit deterministic function growing like a multiple of (Formula presented.). ![]() N2 - We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than (Formula presented.), agrees up to generation (Formula presented.) with a regular (Formula presented.)-ary tree, where (Formula presented.) is the essential minimum of the offspring distribution and the random variable (Formula presented.) is strongly concentrated near an explicit deterministic function growing like a multiple of (Formula presented.). T1 - Galton-Watson trees with vanishing martingale limit ![]()
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