Abstract

We present an elementary proof establishing equality of the right and left-sided $\sqrt{\kappa }$-quantum boundary lengths of an SLE${}_{\kappa }$ curve, $\kappa \in (0,4]$. We achieve this by demonstrating that the $\sqrt{\kappa }$-quantum length is equal to the $(\sqrt{\kappa }∕2)$-Gaussian multiplicative chaos with reference measure given by half the conformal Minkowski content of the curve, multiplied by $2{(4-\kappa )}^{-1}{(1-\kappa ∕8)}^{-1}$ for $\kappa \in (0,4)$ and by $2$ for $\kappa =4$. Our proof relies on a novel “one-sided” approximation of the conformal Minkowski content, which is compatible with the conformal change of coordinates formula.

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