57 problems

The level lines of the ${\mathbb Z\textsc{gff}}$ measure near the corner of the limit shape have fluctuations on the scale $O(L^{1/2})$. Brownian scaling conjecture. Their scaling…

Buser's eigenvalue-sequence conjecture. There exist sequences of closed hyperbolic surfaces whose first eigenvalue tends to $\frac14$.

Consider perturbed simply attractive potentials, including anisotropic potentials, meaning potentials that are not necessarily isotropic. Let the variational principle be the relat…

Assume $d\geq3$ and $m=1$. A rough phase is a phase with the rough, fluctuation-dominated behavior considered in the source. No-rough-phase conjecture. There are no rough phases fo…

Let $U_{\Phi}$ be the set of interior slopes. A non-$\mathcal L$-invariant gradient Gibbs measure $\mu\in\mathcal P(\Omega,\mathcal F^{\tau})$ has approximate slope $u$ if, for eve…

Let $\phi$ be a height function on $\mathbb Z^2$, let $\overline{\phi}$ be its continuous piecewise-linear interpolation on simplices, and define the level sets…

Assume $m=1$ and $d=2$, and consider a general simply attractive model in a rough phase. In the reference examples, suitably rescaled height functions have a scaling limit given by…

Assume $m=1$, $d=2$, and let $U_{\Phi}$ be the set of interior slopes for a simply attractive potential. Let $\sigma$ be the surface tension. Cusp conjecture. The smooth phases wit…

Assume $m=1$, $d=2$, and that $\Phi$ is a general discrete simply attractive potential. Let $\sigma$ denote the surface tension, and let the slopes in the dual of $\mathcal L$ be t…

Let $U_{\Phi}=\{u:|u_1|+|u_2|<\frac{1}{2}\}$ be the set of interior slopes, and let $\mu_u$ be the explicitly described ergodic gradient Gibbs measure of slope $u$. Uniqueness conj…

The $O(n_1,n_2)$ model is the version of the fully packed loop model in which the type $b$ vertex is forbidden. Its height description uses two independent height directions, corre…

A meander is a configuration of a closed non-self-intersecting curve crossing a line at finitely many points, and a semi-meander is the analogous configuration with the line replac…

Let $\mathcal{R}$ and $\mathcal{R}'$ be two surfaces in $\mathcal{S}$. Suppose the cell of $\mathcal{R}$ at position $(i,j)$ has cross ratio $k_{i,j}$, while the cell of…

Let $(X_k)_k$ be a sequence of random finite-area hyperbolic surfaces such that … Write $\lambda_1(X_k)$ for the first positive Laplace eigenvalue. Random hyperbolic-surface spectr…

Let a dimer model be defined on a simply connected bounded domain. At each location in the domain, consider a torus dimer model with weights $(a_0,a_1,a_2)$ chosen according to the…

Fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $g_n/n\rightarrow\theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces. The separatin…

Let $X_g$ be a random closed hyperbolic surface of genus $g$ in one of the Brooks–Makover, Weil–Petersson, or covering models, and let $\lambda_1(X_g)$ denote its first Laplacian e…

Let $\Sigma_g$ be a compact surface of genus $g\geq 0$, with structure group ${\mathrm{SU}}(N)$. Consider its Yang–Mills partition function and Wilson loop expectations. Gross–Tayl…

Let a dimer model have a scaling limit whose t-surface has cusp locations, and let the model's global fluctuations contain a discrete Gaussian component with a shift. Cusp-location…

Large-genus Brownian-surface geodesic length spectrum conjecture. As $g\to\infty$, the random point process $c\,\Xi_g$ converges in distribution to $\mathcal{P}$:

In the $q^{\mathrm{Volume}}$ lozenge tiling model, let the height function be the function recording the tiling's local height, and let the fluctuation field denote its centered an…

Universality conjecture. For a very large class of “higher genus” dimer models, the integral formula for height moments has the same form as the formula given in the paper's Coroll…

Conformal-coordinate conjecture. For more general boundary conditions and other periodic lattices, an appropriate analogue of the critical point mapping gives the correct coordinat…

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathbb{X}'$ be an independent copy of $\mathbb{X}$. A neighborhood of $\mathbb{X}$ means a neighborhoo…

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathscr{G}$ be the graph whose vertices are the faces of $\mathbb{X}$, with an edge between two vertic…