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A Glimpse of the Conformal Structure of Random Planar Maps

Abstract : We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm–Loewner evolution) process of parameter κ=6 and to combine the locality property of the SLE6 together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension 13 almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).
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Submitted on : Wednesday, March 4, 2015 - 3:02:59 PM
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Nicolas Curien. A Glimpse of the Conformal Structure of Random Planar Maps. Communications in Mathematical Physics, Springer Verlag, 2014, 333 (3), pp.1417-1463. ⟨10.1007/s00220-014-2196-5⟩. ⟨hal-01122764⟩



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