An elementary approach to quantum length of SLE

TL;DR

The paper proves an elementary, self-contained result: for SLE$_{\kappa}$ with $\kappa\in(0,4]$ and an independent free-boundary GFF $\Gamma$, the left- and right-hand $\sqrt{\kappa}$-quantum lengths of the SLE curve coincide, and each is represented as a Gaussian multiplicative chaos (GMC) with reference measure given by a one-sided conformal Minkowski content. The authors construct and analyze one-sided Minkowski contents $m_\eta^L$ and $m_\eta^R$, show their equality to the standard content $m_\eta$ up to an explicit constant, and establish a robust link between GMC with base $m_\eta$ and GMC on the Loewner image with Lebesgue reference via the Liouville change-of-coordinates formula. A central technical advance is an elementary, symmetric one-sided approximation and a careful continuity analysis of Liouville measures in both the reference measure and the chaos parameter, allowing a clean passage to subcritical ($\gamma<2$) and critical ($\gamma=2$) regimes. The main results provide explicit constants: the left/right quantum length corresponds to a factor $\frac{2}{(4-\kappa)(1-\kappa/8)}$ times the boundary chaos on the image, and for the critical case $\kappa=4$, $\gamma=2$ the same relation holds with the critical chaos. This work clarifies the structure of quantum length in LQG-SLE settings and supplies a direct, elementary route to the identity of left and right quantum lengths, with implications for conformal welding and related integrable structures.

Abstract

We present an elementary proof establishing the equality of the right and left-sided $\sqrtκ$-quantum lengths for an SLE$_κ$ curve, where $κ\in (0,4]$. We achieve this by demonstrating that the$\sqrtκ$-quantum length is equal to the $(\sqrtκ/2)$-Gaussian multiplicative chaos with reference measure given by half the conformal Minkowski content of the curve, multiplied by $2/(4-κ)$ for $κ\in (0,4)$ and by $1$ for $κ=4$. Our proof relies on a novel "one-sided" approximation of the conformal Minkowski content, which is compatible with the conformal change of coordinates formula.

Figures (3)

• Figure 1: We demonstrate that the $\gamma$-Gaussian multiplicative chaos measures for the fields $\Gamma$ of $f_{T}(\eta([r,s])\cap \mathbb{R}^+)$ and $f_{T}(\eta([r,s])\cap \mathbb{R}^-)$ coincide. Moreover, both measures are equal to a constant times the $\gamma/2$-Gaussian multiplicative chaos measure of $\eta([r,s])$, with the reference measure provided by the conformal Minkowski content of $\eta$ for the field $\Gamma^T$.
• Figure 2: The regions $N_u$ and $f_T(N_u)$ are shaded in grey. The strategy for the proof of Proposition \ref{['p:mainsub']} is to define appropriate measure $\pi_\eta^{\delta,L}$ and $\pi_\eta^{\delta,R}$ supported on the left and right of $\eta$, that both converge to $m_\eta$. By continuity of Gaussian multiplicative chaos with respect to the base measure we have the convergence of $\mu_\Gamma^{{\gamma/2}}[\pi_\eta^{\delta,q}]\to \mu_\Gamma^{{\gamma/2}}[m_\eta]$ for $q=\{L,R\}$. This is illustrated in blue on the left portion of the figure. We then show that the images of the measures $\mu_\Gamma^{{\gamma/2}}[\pi_\eta^{\delta,q}]$ under $(f_T)*$ converge to chaos with respect to Lebesgue measure on the real line (restricted to $\mathbb{R}^-$ when $q=L$ and $\mathbb{R}^+$ when $q=R$).
• Figure 3: Representation of the notation used in the proof. In blue is the curve $\eta$ run up to the first time it hits the ball of center $z$ and radius $r$, and in green is the curve run from this time until the first time that it hits $l$. The left figure illustrates the event $\mathcal{B}_r^s(z,w)$ while the right illustrates the event $\mathcal{U}_r^s(z,w)$.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Free boundary GFF
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• Remark 2.5
• proof
• Definition 2.6
• Definition 2.7: SLE$_\kappa$ Green's function