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A Glimpse of the Conformal Structure of Random Planar Maps

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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 \({ \kappa = 6}\) 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 \({\frac{1}{3}}\) 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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  1. CNRS, Université Paris 6, Paris, France

    Nicolas Curien

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  1. Nicolas Curien
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Correspondence to Nicolas Curien.

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Communicated by F. Toninelli

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Curien, N. A Glimpse of the Conformal Structure of Random Planar Maps. Commun. Math. Phys. 333, 1417–1463 (2015). https://doi.org/10.1007/s00220-014-2196-5

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  • DOI: https://doi.org/10.1007/s00220-014-2196-5

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