Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps




Berestycki, Nathanaël
Laslier, Benoît
Ray, Gourab

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Communications in Mathematical Physics


In this paper we consider random planar maps weighted by the self-dual Fortuin-Kasteleyn model with parameter q is an element of(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 4/pi arccos (root 2-root q/2)= k'/8, where k' 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 is an element of(0,4) . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.




Berestycki, N.; Laslier, B.; & Ray, G. (2017). Critical exponents on Fortuin- Kasteleyn weighted planar maps. Communications in Mathematical Physics, 355(2), 427-462. DOI: 10.1007/s00220-017-2933-7