70 problems

Bogomolny–Schmit conjecture. Their nodal lines at large scales should converge to curves described by Schramm–Loewner evolution with parameter $6$, namely $\operatorname{SLE}_6$.

In two dimensions, let self-avoiding walk be rescaled in the usual way, and let $\operatorname{SLE}_{8/3}$ denote Schramm–Loewner evolution with parameter $8/3$. Lawler–Schramm–Wer…

Wieland–Wilson's conjecture. The variance of the winding angle satisfies

Let $\mathfrak{h}$ be a unit-area $1$-LQG sphere independent of the three whole-plane space-filling curves $(\eta^{\mathrm{b}},\eta^{\mathrm{r}},\eta^{\mathrm{g}})$ obtained from t…

Aizenman's conformal invariance conjecture. The scaling limits of critical percolation interfaces are conformally invariant and their crossing probabilities satisfy Cardy's formula…

Scaling-limit conjecture. As $\delta\to0$, the measures ${\bf P}_{D,\delta}$ converge to a probability measure ${\bf P}_D$ on simple paths from the origin to $\partial D$, with

Consider independent Bernoulli percolation with parameter $p=1/2$ on the faces of a gasket obtained from a critical $O(n)$ model, whose scaling limit is expected to be a conformal…

Let $\gamma$ be a path sampled from the relevant SLE process, and define its trunk by the set of points where the driving function and force-point process agree. Let $\beta$ be the…

Nienhuis–Kager conjecture. The scaling limit of $P$ is $\mathrm{SLE}_\kappa$ with $n=-2\cos(4\pi/\kappa)$ and $4\leq\kappa\leq 8$ if $0<n\leq2$ and $x>x_c$; it is…

A self-avoiding random walk is a lattice path that visits no lattice site more than once. Its scaling limit means the limiting continuum law obtained after rescaling the lattice sp…

Consider the double domino interface in a domino tiling, and let the lattice mesh tend to zero. The associated domino tiling height function is expected to have the Gaussian free f…

Consider the path arising from the double domino tiling model in plane strips. SLE4 scaling-limit conjecture. The scaling limit of this path should be the trace of…

The near-critical FK-Ising model is considered in a regime where its interfaces have a scaling limit, with the critical scaling limit described by Schramm–Loewner evolution with pa…

Let $\mathcal{J}(\boldsymbol{z},u)$ be the screening solutions constructed by choosing integration contours for the screening fields, and let the null vector equations and Ward ide…

Let $\lambda\neq 0$ be fixed. Consider any subsequential scaling limit of FK-Ising percolation at $\beta=\beta_c+\frac{\lambda}{n}$ in $\frac{1}{n}\mathbb{Z}^2\cap[-1,1]^2$, with D…

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).

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…

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…

A quadrangulated disk is a planar map whose interior faces have four edges and whose exterior face has a simple boundary; a self-avoiding bubble is a cyclic sequence of distinct ed…

The biased Brownian separable permuton is denoted by $\bm \mu_{1,q}$ for $q\in[0,1]$. The critical $2$-LQG sphere is the Liouville quantum gravity sphere at parameter $2$. Critical…

LQG–SLE mating-map conjecture. Almost surely,

LQG–SLE skew Brownian SDE conjecture. The process satisfies

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…