Brownian motion correlation in the peanosphere for $\kappa > 8$

TL;DR

This paper extends the known correlation formula for Brownian motion in the peanosphere construction to all  > 4, enabling convergence proofs for  > 8 in Liouville quantum gravity decorated with SLE.

Contribution

It proves that the Brownian motion correlation formula  > 8 matches the previously known formula for 4 <   8, filling a key gap in the theory.

Findings

Correlation formula  > 8 confirmed as -33 for the Brownian motion.

Established the link between tail exponents of SLE on quantum wedges and Brownian motion.

Enabled convergence results for  > 8 in LQG decorated with SLE.

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.