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Geometry of large random planar maps with a prescribed degree sequence

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RGMW06 - RGM follow up

I will discuss some recent progress and still ongoing work about the scaling limit of the following configuration-like model on random planar maps: for every integer n, we are given n deterministic (even) integers and we sample a planar map uniformly at random amongst those maps with n faces and these prescribed degrees. Under a `no macroscopic face' assumption, these maps converge in distribution after suitable scaling towards the celebrated Brownian map, in the Gromov-Hausdorff-Prokhorov sense. This model covers that of p-angulations when all the integers are equal to some p, which we can allow to vary with n, without constraint; it also applies to so-called Boltzmann random maps and yields a CLT for planar maps.




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