29 problems

An observable for SLE$_2$ is a process associated with the evolving SLE domain that is a martingale; in the examples discussed, such observables include positive harmonic functions…

Self-avoiding-walk SLE conjecture. If the scaling limit of self-avoiding walk exists and is conformally invariant, then its only possible limit is $SLE_{8/3}$; the source further r…

SLE dimension conjecture. The Hausdorff dimension of $\gamma[0,t]$ for chordal or radial $SLE_\kappa$ is conjectured to be

Self-avoiding-walk dimension conjecture. It is conjectured that the scaling limit of planar self-avoiding walks has paths of dimension $4/3$.

Rohde–Schramm's conjecture. The random cluster representation of the Potts model for $0<q<4$ is related to the SLE process.

The SLE conjecture. For $\kappa=8/3$, the SLE trace gives the scaling limit of the SAW restricted to the half plane.

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 $G_n$ be the partially wired square grid described in the source, let $\omega$ have the critical random-cluster measure with $q\in(0,4)$, let $\omega^\dagger$ be its dual confi…

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 the UST Peano curve be the interface associated with a uniform spanning tree and its dual on a planar lattice, and let $\textrm{SLE}_8$ denote stochastic Loewner evolution with…

Let $\gamma$ be the trace of chordal $\textrm{SLE}_\kappa$. SLE trace dimension conjecture. When $\kappa<8$, the Hausdorff dimension of $\gamma$ is almost surely … The source gives…

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…

Let $K_1$ be the time-one hull of chordal $\textrm{SLE}_\kappa$ in the upper half-plane, and let $\partial K_1$ denote its outer boundary. Kenyon's outer-boundary dimension conject…

Let chordal $\textrm{SLE}_\kappa$ denote the random curve generated by chordal stochastic Loewner evolution with parameter $\kappa\geq 0$, and let the UST Peano curve be the interf…

Subtraction conjecture. The subtractions in the corresponding evolution equation take this loss into account.

Continuous-transfer-operator conjecture. The continuous version of $\widetilde T$ is, more precisely,

The $O(n)$ conformal field theory contains boundary one-leg operators $\phi_a(Re^{i\theta})$ carrying an $O(n)$ index $a$, together with bulk $N$-leg operators, and the joint proba…

Let $\mathrm{SLE}^{\mathbb H}_{\kappa,(0;0^-,0^+)\to\varepsilon}(\rho_-,\rho_+)$ denote the chordal SLE process in $\mathbb H$ with two force points at $0^-$ and $0^+$, taken in th…

Let $L_{\mathfrak{sl}_{n},1}$ be the basic representation of $\widehat{\mathfrak{sl}}_{n}$, and let $X_{1},\ldots,X_{n^{2}-1}$ be an orthonormal basis of $\mathfrak{sl}_{n}$. Let…

The paper considers SLE-type growth processes associated with null vectors in basic representations of affine Lie algebras. In particular, such processes are constructed for the ba…

Kontsevich–Suhov uniqueness conjecture. For $c\leq 1$ and $h=h_{2,1}(\tau)$, such an assignment is unique up to multiplication by a constant.

Consider the nodal lines of the two-dimensional Gaussian random wave model and of wave functions of chaotic billiards. SLE conjecture. The nodal lines for such systems are a realis…

Beta-spectrum conjecture. The equation

Let critical site percolation be defined on the triangular lattice, and consider the boundary or exploration interface of a critical percolation cluster. The SLE6 scaling-limit con…

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…