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 annealed critical exponent associated with the length of cluster interfaces, which is shown to be
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.
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Communicated by H.-T. Yau
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.
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Berestycki, N., Laslier, B. & Ray, G. Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps. Commun. Math. Phys. 355, 427–462 (2017). https://doi.org/10.1007/s00220-017-2933-7
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DOI: https://doi.org/10.1007/s00220-017-2933-7