Growth-fragmentations, Brownian cone excursions and SLE(6) explorations of a quantum disc

TL;DR

The paper reveals a growth-fragmentation structure embedded in Brownian cone excursions of angle $2\pi/3$, proving that the resulting fragment process $\mathbf{Z}$ is governed by a positive self-similar Markov process with index $\tfrac{3}{2}$, i.e., $X^{3/2}$, via a Lamperti-type time change. In the Brownian framework (no reliance on SLE tools beyond the mating-of-trees encoding), the authors establish a detailed excursion theory for forward and backward cone times, including a Bismut description and an explicit Lévy structure for the associated fragmentation, and connect these findings to the $\gamma=\sqrt{8/3}$ Liouville quantum gravity disc coupled to space-filling $SLE_6$ explorations. The results yield precise laws for the duration and start points of cone excursions, a target-invariance property, and a pathwise construction of the $3/2$-stable process conditioned to stay positive, all framed within the growth-fragmentation perspective. The core contribution is the elementary Brownian approach to the Brownian-LQG-SLE correspondence in this setting, culminating in a complete description of the uniform exploration, the locally largest fragment $Z^*$, and the martingale convergence towards the excursion duration, thereby strengthening the bridge between Brownian geometry, self-similar processes, and quantum gravity explorations.

Abstract

The aim of this article is to present a growth-fragmentation process naturally embedded in a Brownian excursion from boundary to apex in a cone of angle $2π/3$. This growth-fragmentation process corresponds, via the so-called mating-of-trees encoding arXiv:1409.7055, to the quantum boundary length process associated with a branching $\mathrm{SLE}_6$ exploration of a $γ= \sqrt{8/3}$ quantum disc. However, our proof uses only Brownian motion techniques, and along the way we discover various properties of Brownian cone excursions and their connections with stable Lévy processes. Assuming the mating of trees encoding, our results imply several fundamental properties of the $γ= \sqrt{8/3}$-quantum disc $\mathrm{SLE}_6$-exploration.

Figures (10)

• Figure 1: The growth-fragmentation $\mathbf{Z}$ embedded in cone excursions. If $t$ is a time in the excursion, we record the forward cone excursions of $e^{t,-}$ (blue). We also depict some nested cone excursions in grey, to suggest that there is an accumulation of them inside each maximal excursion. For $0\le {b} \le {\varsigma^t}$, we construct (purple) an interval $(g^t(b),d^t(b))$ containing $t$ such that $g^t(b)=t-\uptau^t(b)$ and $d^t(b)-t$ is the first simultaneous running infimum (red) of the path $e^{t,+}$ run from time $t$ to time $\zeta$, that also falls below the whole trajectory from $g^t(b)$ to $t$.
• Figure 2: A unit boundary length quantum disc decorated with an independent space-filling $\mathrm{SLE}_{\kappa'}$$\eta$ from $-i$ to $-i$, parametrised by quantum area. $L_t$ corresponds to the quantum length of the blue curve. $R_t$ corresponds to one plus the quantum length of the green curve minus the quantum length of the red one.
• Figure 3: Branches of the space-filling $\mathrm{SLE}_6$$\eta$ on the $\sqrt{8/3}$--quantum disc towards $x$ and $y$. (a) The branch of $\eta$ towards $x$ and $y$ (purple) is the same. (b) The two branches get disconnected: a loop has been cut out, surrounding $x$. The branch $\eta^x$ targeted at $x$ is shown in (purple and then) red, and the branch $\eta^y$ targeted at $y$ is in (purple and then) blue.
• Figure 4: The total boundary process towards $z\in\mathbb{D}$. At time $s$ along the branch $\eta^z$, we record the total boundary length (dashed red) of the component containing $z$ (blue).
• Figure 5: Different situations when the branch towards $z$ has a jump in MSW: (a) the CPI discovers a new $\mathrm{CLE}_{\kappa}$ loop; (b) the CPI hits the boundary of $\mathbb{D}$ (or itself); (c) the CPI hits a previously visited $\mathrm{CLE}_{\kappa}$ loop. The first case corresponds to positive jumps, (b) and (c) to negative jumps.

Theorems & Definitions (78)

• Theorem 1.1: Growth-fragmentation process: cone excursions
• Theorem 1.2: Growth-fragmentation: $\sqrt{8/3}$--LQG
• Proposition 1.3: Law of duration conditioned on displacement
• Corollary 1.4: Law of area of unit-boundary quantum disc
• Proposition 1.5: Target-invariance: cone excursions
• Corollary 1.6: Target-invariance: LQG
• Proposition 1.7: Law of the uniform exploration: cone excursions
• Corollary 1.8: Law of the uniform exploration: LQG
• Proposition 1.9: Pathwise construction of the spectrally positive $\frac{3}{2}$--stable process conditioned to stay positive -- informal version
• Theorem 1.10