Harmonic measure and SLE

Convergence to closed-form distribution for the backward $SLE_κ$ at some random times and the phase transition at $κ=8$

We study a one-dimensional SDE that we obtain by performing a random time change of the backward Loewner dynamics in $\mathbb{H}$. The stationary measure for this SDE has a closed-form expression. We show the convergence towards its stationary measure for this SDE, in the sense of random ergodic averages. The precise formula of the density of the stationary law gives a phase transition at the value $κ=8$ from integrability to non-integrability, that happens at the same value of $κ$ as the change in behavior of the $SLE_κ$ trace from non-space filling to space-filling curve. Using convergence i…

Large deviations of radial SLE$_{\infty}$

We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname{SLE}$) on the unit disk with parameter $κ\rightarrow \infty$. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{ϕ_t^2 (ζ)\, dζ\}_{t \in [0,1]}$ on the unit circle and equals $\int_0^1 \int_{S^1} |ϕ_t'|^2/2\,dζ\,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.

$\text{SLE}_κ$ 环路测度的 Onsager–Machlup 泛函

Onsager-Machlup functional for $\text{SLE}_κ$ loop measures

本文建立 Brown 环路测度两种重整化的联系，并据此将 Loewner 能量表述为 $\text{SLE}_κ$ 及同中心荷 MKS 环路测度的 Onsager–Machlup 泛函。

通过量子曲面的共形焊接得到的SLE气泡测度

The SLE Bubble Measure via Conformal Welding of Quantum Surfaces

本文证明SLE气泡测度自然产生于两个Liouville量子引力圆盘的共形焊接，并给出分解公式，进而计算SLE气泡共形半径的矩，并将Liouville共形场论中的体-边界关联函数与量子边界长度和量子面积联系起来。

A continuous proof of the existence of the SLE$_8$ curve

Suppose that $η$ is a whole-plane space-filling SLE$_κ$ for $κ\in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $η(0) = 0$. For each $T > 0$ and $κ\in (4,8)$ we let $μ_{κ,T}$ denote the law of $η|_{[0,T]}$. We show for each $ν, T > 0$ that the family of laws $μ_{κ,T}$ for $κ\in [4+ν,8)$ is compact in the weak topology associated with the space of probability measures on continuous curves $[0,T] \to {\mathbf C}$ equipped with the uniform distance. As a direct byproduct of this tightness result (taking a limit as $κ\uparrow 8$), we obtain a new proof o…

Schramm Loewner Evolution and Liouville Quantum Gravity

We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.

SLE环测度的共形场论与Kontsevich–Suhov猜想

The CFT of SLE loop measures and the Kontsevich--Suhov conjecture

本文研究SLE_κ环测度的共形场论，构造Virasoro代数的可交换表示，证明Kac表奇异向量存在性等，首次证明Kontsevich–Suhov猜想的限制测度唯一性，为后续研究奠定基础。

Characterizations of SLE$_κ$ for $κ\in (4,8)$ on Liouville quantum gravity

We prove that SLE$_κ$ for $κ\in (4,8)$ on an independent $γ=4/\sqrtκ$-Liouville quantum gravity (LQG) surface is uniquely characterized by the form of its LQG boundary length process and the form of the conditional law of the unexplored quantum surface given the explored curve-decorated quantum surface up to each time $t$. We prove variants of this characterization for both whole-plane space-filling SLE$_κ$ on an infinite-volume LQG surface and for chordal SLE$_κ$ on a finite-volume LQG surface with boundary. Using the equivalence of Brownian and $\sqrt{8/3}$-LQG surfaces, we deduce that SLE$_…