The boundary of random planar maps via looptrees

We study the scaling limits of the boundary of Boltzmann planar maps conditioned on having a large perimeter. 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 α∈(1,2). In the so-called dense phase α∈(1,3/2), it was established by Richier that the scaling limit of the boundary is a stable looptree. In this work, we complete the picture by proving that in the dilute phase α∈(3/2,2) (as well as in the generic critical regime), the scaling limit is a multiple of the unit circle. 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. The subcritical regime is also investigated. In this case, we show that the scaling limit of the boundary is a multiple of the Brownian CRT instead. The strategy consists in studying scaling limits of looptrees associated with specific Bienaymé--Galton--Watson trees. In the first case, it relies on an invariance principle for random walks with negative drift, which is of independent interest. In the second case, we obtain the more general result that the Brownian CRT is the scaling limit of looptrees associated with BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution, confirming thereby a prediction of Curien and Kortchemski