19 problems

Let $a=\log q$ for $q\in(0,1)$, and let $\mathbb{P}(N)(q)$ denote the probability of the annular crossing event $N$ described above. Set … Here $\eta$ denotes Dedekind's eta functi…

Let $\mathcal{Z}_{\alpha}(\boldsymbol{\theta})$ and $\mathcal{Z}_{\alpha}^{\eta}(\boldsymbol{\theta})$ denote the zero-spin and spin-$\eta$ pure partition functions, respectively,…

Let $\mathrm{LP}(n,m)$ be the set of radial link patterns, let $\mathcal{Z}_{\alpha}(\boldsymbol{\theta})$ be the pure partition functions, let…

Beltrami-welding conjecture. The above Beltrami welding yields an SLE loop measure and thus an independent second construction of the SLE loop in Theorem 1.3 of Ang (2021).

Fix $W_1\geq\frac{\gamma^2}{2}$ and $W>2$. Let $\mathrm{Bubble}_{\mathbb H}(p)$ denote the space of bubbles in $\mathbb H$ rooted at $p$, and let…

LQG–SLE mating-map conjecture. Almost surely,

Let $\Lambda_n$ be the finite domain, let $T$ be sufficiently small, and let $\psi_n$ be a Villain model in $\Lambda_n$ at temperature $T$ with boundary values $\exp(-ia)$ on…

Let $\Lambda_n$ be the finite domain, let $T$ be sufficiently small, and let $\psi_n$ be a Villain model in $\Lambda_n$ at temperature $T$. Give it boundary values…

Let $\phi_n$ be the discrete Gaussian free field, let $u_n$ be the auxiliary function appearing in the construction, and let $\eta^{(n)}$ be the original level line. For sufficient…

In the setting of the spanning-tree LQG scaling-limit conjecture, let $\lambda^{\infty,n}$ be the image in $\mathbb C$ of the Peano curve $\lambda^\infty$ of $T^\infty$ under the u…

Fix $\kappa\in(0,4]$, let $n\geq 1$, and let $\alpha_1,\ldots,\alpha_n$ denote the parameters of the restriction measures. Define … The generalized intersection exponents are the e…

Let $\Lambda$ be the triangular, hexagonal, or square lattice. Suppose either $y=0$ and $z>0$, or $y\in[0,1]$ and $z\in[1,\infty)$. Let $T$ be a spanning tree on $\Lambda$ sampled…

Joint interface-function convergence conjecture. The joint law of both NW/SE and NE/SW interface functions of a random bipolar-oriented planar map converges to the joint law of bot…

Let $D$ be a bounded, simply connected domain containing the origin whose boundary is a Jordan curve, and let $U,V$ be disjoint, closed subarcs of $\partial D$. For the lattice app…

Fix $q\in[0,4]$. A critical FK interface is the exploration interface in a Dobrushin domain for FK percolation with cluster-weight $q$. Schramm's conjecture. The law of critical FK…

Fix $n\in[0,2]$ and consider the loop $O(n)$ model with an interface between boundary points $a$ and $b$. Smirnov's O(n)-model interface conjecture. For $z=1/\sqrt{2+\sqrt{2-n}}$,…

Let $\Omega\subsetneq\mathbb{C}$ be a simply connected domain, let $z\in\Omega$, and let $a,b$ be distinct boundary points, with the boundary smooth near $b$. For $\delta>0$, let…

Let $\Omega$ be a simply connected domain in $\mathbb{C}$ with boundary points $a$ and $b$, and let $(\Omega_\delta,a_\delta,b_\delta)$ be discrete approximations. At the critical…

Super-strong separation lemma. Conditioned on at least four of these interfaces reaching $B(R_1)$, the set of interfaces reaching $R_1$ has positive quality with positive probabili…