## 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.