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. Communications in Mathematical Physics Springer Journals

Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps

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Springer Berlin Heidelberg
Copyright © 2017 by The Author(s)
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
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