## Critical Exponents on Fortuin-Kasteleyn Weighted Planar Maps

##### Authors

Berestycki, Nathanael

Laslier, Benoit

Ray, Gourab

##### Publication Date

2017-10-01##### Journal Title

COMMUNICATIONS IN MATHEMATICAL PHYSICS

##### ISSN

0010-3616

##### Publisher

Springer

##### Volume

355

##### Issue

2

##### Pages

427-462

##### Type

Article

##### This Version

AM

##### Metadata

Show full item record##### Citation

Berestycki, N., Laslier, B., & Ray, G. (2017). Critical Exponents on Fortuin-Kasteleyn Weighted Planar Maps. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 355 (2), 427-462. https://doi.org/10.1007/s00220-017-2933-7

##### Abstract

In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be $$ \frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8}. $$ where $\kappa' $ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop which is shown to be 1 for all values of $q \in (0,4)$. Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.
Communicated by H.-T. Yau

##### Sponsorship

Nathanaël Berestycki: Supported in part by EPSRC grants EP/L018896/1 and EP/I03372X/1.
Benoît Laslier: Supported in part by EPSRC grant EP/I03372X/1.
Gourab Ray: Supported in part by EPSRC grant EP/I03372X/1.

##### Funder references

EPSRC (EP/I03372X/1)

EPSRC (EP/L018896/1)

##### Identifiers

External DOI: https://doi.org/10.1007/s00220-017-2933-7

This record's URL: https://www.repository.cam.ac.uk/handle/1810/268296

##### Rights

Attribution 4.0 International, Attribution 4.0 International