Critical Exponents on Fortuin–Kasteleyn Weighted Planar Maps

Date

2017

Authors

Berestycki, Nathanaël
Laslier, Benoît
Ray, Gourab

Journal Title

Journal ISSN

Volume Title

Publisher

Communications in Mathematical Physics

Abstract

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.

Description

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Citation

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