Critical exponents for FK random planar maps

Nathanael Berestycki (University of Cambridge)

Frank Adams 2,

Random surfaces have recently emerged as a subject of central
importance in probability theory.
We consider random planar maps weighted by the self-dual
Fortuin--Kastelyn model with parameter $q \in (0,4)$, which can be
thought of as canonical discretisations of the surface of interest.
The case $q=1$ of percolation corresponds to uniform random planar
maps, about which detailed geometric information is known thanks in
particular to the work of Le Gall, Miermont, and others. In this work
we address the generic case and are able to obtain rigorously the
value of critical exponents associated with the length of cluster
interfaces, which is shown to be
\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right).
This is consistent with physics predictions; in particular, applying
the so-called KPZ formula we recover the dimension of SLE curves. No
prior knowledge in this area will be required, and I will explain how
this result fits in with the global picture of random planar geometry.

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