21 problems

Let $b\in\mathbb{R}$, and define $a=a(b)>0$ by … Set $\kappa=2/a^2$ and let $\kappa'=\kappa(-b)$, so that $\kappa\kappa'=16$. Duplantier's SLE duality conjecture. When $\kappa>4$,…

Let $D\subset\mathbb{C}$ be a simply connected domain, let $x,y$ be distinct boundary points, and let $\eta$ be a chordal curve from $x$ to $y$. A random curve from $x$ to $y$ is r…

Let $b(t)$, $t\in[0,1]$, be a two-dimensional Brownian motion with $b(0)=0$. Its frontier of Brownian motion (FBM) is the boundary of the unbounded connected component of the compl…

Let $D$ be a simply connected domain, let $a,b\in\partial D$ be distinct boundary points, and consider the chordal $\textrm{SLE}_\kappa$ trace from $a$ to $b$, modulo reparameteriz…

Let $\mathbb{U}$ be the unit disk and consider a square grid of mesh $\epsilon$. Let the uniform measure be taken on simple grid paths from $0$ to $\partial\mathbb{U}$, and separat…

Let $\gamma$ be the trace of chordal $\textrm{SLE}_\kappa$ in the upper half-plane. SLE8 trace continuity conjecture. For $\kappa=8$, the trace $\gamma$ is almost surely a continuo…

For $\kappa\in[0,8]$, let $\mathrm{SLE}_\kappa$ denote Schramm–Loewner evolution with parameter $\kappa$, and let its trace be the associated random curve. SLE trace minimality con…

Let $\mathrm{SLE}_\kappa(\underline{\rho})$ be chordal SLE with multiple marked force points, with $\kappa\in(0,8)$, and suppose that the curve is non-boundary filling, meaning it…

Let $\mathrm{SLE}_\kappa$ be chordal Schramm–Loewner evolution in a simply connected domain, viewed as a random curve between two boundary points, and identify curves up to time re…

Natural parameterization conjecture. The natural time parameterization is the time parameterization that arises when ${\rm SLE}$ is considered as the scaling limit of an interface…

Alvisio–Lawler conjecture. There should exist $\beta>0$ such that, on any compact time interval, there almost surely exists a constant $c>0$ for which

Let $C$ be a random real plane curve of degree $d$, and let $h(C)$ denote the energy of its oval arrangement. The expectation with respect to the random-curve model is written as…

Consider one of the loop measures ${\mathbb P}_{LC}$ or ${\mathbb P}_{inc}$, and let $k\geq 1$ denote the number of loops. Superexponential loop-number decay conjecture. There exis…

Let ${\mathbb P}_{LC^0}$ be the limiting loop measure associated with the zero-curvature scaling, and let $\Omega_1$ denote the event that the loop has nonzero area. Macroscopic-lo…

Lawler–Sheffield conjecture. The natural length of an $SLE$ path is given by its $d$-dimensional Minkowski content.

Let $SLE_\kappa$ be a Schramm–Loewner evolution path with Hausdorff dimension … The natural length conjecture. The natural length of an $SLE_\kappa$ path is given by an appropriate…

For \boldsymbol{\text{SLE}}_\boldsymbol{\text{{\kappa}}}, let $boldsymbol{\text{simple path}}$ mean that the hull generated by the Loewner evolution is a simple curve. Schramm's…

The construction produces random closed curves by conformally welding random homeomorphisms of the unit circle. For $\beta<\beta_c$ (in the paper's high-temperature regime,…

For $\operatorname{SLE}(\kappa)$, let $C$ and $D$ denote the bridge heights and bridge lines, respectively, when these objects exist. At $\kappa=0$ and $\kappa=8/3$, their Hausdorf…

In the loop gas and Liouville-gravity descriptions, a bulk operator is understood as creating a closed loop, while an $SLE$-type process describes a random curve. Bulk-loop/SLE cor…

Càdlàg-generation conjecture. The increasing hulls $(K_t)$ are generated by càdlàg curves.