# Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps

Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps In this paper we consider random planar maps weighted by the self-dual Fortuin–Kasteleyn model with parameter $${q \in (0,4)}$$ 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 annealed critical exponent associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8},$$ 4 π arccos 2 - q 2 = κ ′ 8 , where $${\kappa' }$$ κ ′ 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 $${q \in (0,4)}$$ q ∈ ( 0 , 4 ) . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

# Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps

, Volume 355 (2) – Jul 22, 2017
36 pages
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by The Author(s)
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-2933-7
Publisher site
See Article on Publisher Site

### Abstract

In this paper we consider random planar maps weighted by the self-dual Fortuin–Kasteleyn model with parameter $${q \in (0,4)}$$ 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 annealed critical exponent associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8},$$ 4 π arccos 2 - q 2 = κ ′ 8 , where $${\kappa' }$$ κ ′ 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 $${q \in (0,4)}$$ q ∈ ( 0 , 4 ) . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.

### Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Jul 22, 2017

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