# Brownian motion correlation in the peanosphere for κ &gt;8

 dc.creator Gwynne, E dc.creator Holden, N dc.creator Miller, Jason Peter dc.creator Sun, X dc.date.accessioned 2016-06-20 dc.date.accessioned 2018-11-24T23:27:33Z dc.date.available 2018-01-19T11:19:26Z dc.date.available 2018-11-24T23:27:33Z dc.date.issued 2017-11-01 dc.identifier https://www.repository.cam.ac.uk/handle/1810/270827 dc.identifier.uri http://repository.aust.edu.ng/xmlui/handle/123456789/4003 dc.description.abstract The peanosphere (or "mating of trees") construction of Duplantier, Miller, and Sheffield encodes certain types of $\gamma$-Liouville quantum gravity (LQG) surfaces ($\gamma \in (0,2)$) decorated with an independent SLE$_{\kappa}$ ($\kappa = 16/\gamma^2 > 4$) in terms of a correlated two-dimensional Brownian motion and provides a framework for showing that random planar maps decorated with statistical physics models converge to LQG decorated with an SLE. Previously, the correlation for the Brownian motion was only explicitly identified as $-\cos(4\pi/\kappa)$ for $\kappa \in (4,8]$ and unknown for $\kappa > 8$. The main result of this work is that this formula holds for all $\kappa > 4$. This supplies the missing ingredient for proving convergence results of the aforementioned type for $\kappa > 8$. Our proof is based on the calculation of a certain tail exponent for SLE$_{\kappa}$ on a quantum wedge and then matching it with an exponent which is well-known for Brownian motion. dc.publisher Elsevier dc.publisher Annales de l'institut Henri Poincare (B) Probability and Statistics dc.title Brownian motion correlation in the peanosphere for κ >8 dc.type Article
﻿

## Files in this item

FilesSizeFormatView
covariance-matrix-final.pdf479.2Kbapplication/pdfView/Open