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Some scaling limit results for critical Fortuin-Kastelyn random planar map model

Date: 
Thursday 29th January 2015 - 13:30 to 14:30
Venue: 
INI Seminar Room 1
Abstract: 
Co-authors: Ewain Gwynne (MIT), Cheng Mao (MIT)

Sheffield (2011) introduced a discrete inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model and showed that a certain two-dimensional random walk associated with an in finite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion. We improve on this scaling limit result by showing that the times corresponding to FK loops (or "flexible orders") in the discrete model converge to the \pi/2 cone times of the Brownian motion. Our result can be used to obtain convergence of interesting functionals of the FK loops including their lengths and areas toward the corresponding "quantum" functionals of the loops of a conformal loop ensemble on a Liouville quantum gravity surface, hence provides a solution to the whole plane version of a question by Duplantier, Miller, and Sheffield (2014).

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons