math.PR

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Title: Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times

Abstract: Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. He showed that a certain two-dimensional random walk associated with an infinite-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 inventory accumulation model converge in the scaling limit to the $\pi/2$-cone times of the correlated Brownian motion. This statement implies a scaling limit result for the joint law of the areas and boundary lengths of the bounded complementary connected components of the FK loops on the infinite-volume planar map. In light of the encoding of Duplantier, Miller, and Sheffield (2014), the limiting object coincides with the joint law of the areas and boundary lengths of the bounded complementary connected components of a collection of CLE$_\kappa$ loops on an independent Liouville quantum gravity cone.
 Comments: 58 pages, 8 figures, version 3 contains a new section which explicitly proves a scaling limit result for areas and boundary lengths of components of FK loops Subjects: Probability (math.PR); Mathematical Physics (math-ph) Cite as: arXiv:1502.00546 [math.PR] (or arXiv:1502.00546v3 [math.PR] for this version)

Submission history

From: Ewain Gwynne [view email]
[v1] Mon, 2 Feb 2015 16:59:50 GMT (215kb,D)
[v2] Mon, 4 May 2015 15:30:27 GMT (218kb,D)
[v3] Mon, 8 Aug 2016 17:24:19 GMT (364kb,D)
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