21 problems

Let $\mathcal{Z}_{\alpha}(\boldsymbol{z},u)$ and $\mathcal{Z}_{\alpha}(\boldsymbol{x})$ be pure partition functions, and let $\Phi(\boldsymbol{x},\boldsymbol{\xi},u)$ and…

Whole-plane SLE reversibility conjecture. The whole-plane SLE$_\kappa$ trace satisfies reversibility for $\kappa\in(0,4]$.

Let $\kappa>4$ and set $\kappa'=16/\kappa$. In the configuration $(\mathbb{H},x,y,z,\infty)$ with $x<y<z$, let $(K_t)$ be the Loewner chain obtained by running an…

Dubédat's boundary duality conjecture. The right boundary of $SLE^\sigma(\kappa,\kappa-4)$ is an $SLE(\kappa',(\kappa'-4)/2)$. The conjecture is a precise proposed realization of S…

Let $\mathbb H$ be the upper half-plane and let $\gamma$ be a chordal $SLE_\kappa$ path from the origin to infinity, with fixed $\kappa\leq 8$. SLE reversibility conjecture. The la…

Let $\gamma$ denote a rescaled self-avoiding curve, and let $SLE_{8/3}$ denote chordal Schramm–Loewner Evolution with parameter $8/3$. Self-avoiding-walk scaling-limit conjecture.…

For each $\underline{\kappa}\in(0,4)$, let $\rho_c=\rho_c(\underline{\kappa})$ lie in $(-2-\underline{\kappa}/2,-2)\cap[\underline{\kappa}/2-4,-2)$, and let $\eta$ be an SLE…

For $\kappa \in (4,8)$, let $\eta$ be an SLE$_\kappa$ curve from $0$ to $\infty$ in $\mathbb{H}$. The adjacency graph of the connected components of $\mathbb{H} \setminus \eta$ has…

Fix irrational $\kappa\in(0,8)$ and let $\mathrm{LP}(n,m)$ be the set of link patterns. Let $\mathcal{Z}_{\alpha}$ be the pure partition function indexed by…

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…

Radial SLE scaling-limit conjecture. There exist $b$, $\tilde b_n$, a critical value $\beta=\beta_{\cent}$, and a partition function $\Psi^*(D;{\bf z},0)$ such that, as…

Let $\kappa<8$, and consider the backward $SLE_{\kappa}$ flow and its $SLE_{\kappa}$ trace. At any fixed deterministic time $t>0$, let the tip of the trace be viewed through the co…

Critical random-cluster interface conjecture. As $\delta\to 0$, the interface in $(\Omega^{\delta};x^{\delta},y^{\delta})$ converges weakly to the chordal…

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

Let $\kappa\in(0,8)$, let $\nu>(-4)\vee(\kappa/2-6)$, and let $(\Omega;x_1,x_2,x_3,x_4)$ be a quad. Let $\eta$ be an $\operatorname{hSLE}_{\kappa}(\nu)$ in $\Omega$ from $x_1$ to…

Let $\tilde{\Omega}$ be the space of equivalence classes of SLE(2) curves in $\mathbb{D}$, let $\mathcal{M}$ be the relevant space of measures, and let $G(z)=G_{\mathbb{D}}(z)$ den…

Self-avoiding-walk SLE conjecture. For $x=x_c$, the law of $\gamma_\delta$ in $(\Omega_\delta,a_\delta,b_\delta)$ converges as $\delta\to0$ to chordal Schramm–Loewner Evolution wit…

Reversibility conjecture. After a time-change, $(W_0(\beta_0(1/t)))$, $0<t<\infty$, has the same distribution as $(\beta_0(t))$, $0<t<\infty$.

Reversibility of chordal SLE with force points. After a time-change, the reversal of $(W(\beta(t)))$ has the same distribution as a chordal SLE$(\kappa;\rho_-,\rho_+)$ trace starte…

Let $\bar\beta(t)$ denote the average spectrum of SLE, and let $t_{\min}$ and $t_{\max}$ be the two points such that a tangent to $\bar\beta(t)$ intersects the $y$-axis at $-1$. The…

Let $q\in[0,4]$, and consider the Fortuin–Kasteleyn random-cluster model on the square lattice at the self-dual value $p_{\mathrm{sd}}=\sqrt q/(\sqrt q+1)$, with Dobrushin boundary…