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A glimpse of the conformal structure of random planar maps · Around

2013

A glimpse of the conformal structure of random planar maps

Curien, Nicolas

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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 SLE_{6} 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.

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