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Critical Exponents on Fortuin-Kasteleyn Weighted Planar Maps

Accepted version
Peer-reviewed

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Authors

Berestycki, Nathanael 
Laslier, Benoit 
Ray, Gourab 

Abstract

In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter q∈(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∈(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

Description

Keywords

math.PR, math.PR, math-ph, math.MP, 60K35, 60J67, 60D05

Journal Title

COMMUNICATIONS IN MATHEMATICAL PHYSICS

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

355

Publisher

Springer
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)
Engineering and Physical Sciences Research Council (EP/L018896/1)
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.