92 problems

Gaussian free field–discrete Gaussian conjecture. The fluctuations of the dimer-model height function in the liquid region of a graph of scale $N$ are approximated in distribution…

Kenyon–Okounkov conjecture. In the interior of $\mathcal{L}$, the fluctuations are governed by the pullback of the Gaussian free field from $\mathbb{H}$.

Ashkin–Teller magnetisation-field decomposition conjecture. The statement of the main magnetisation-field decomposition theorem holds for the Ashkin–Teller magnetisation field at e…

Kenyon–Okounkov conjecture. The limit of the height fluctuations of $h_{Omega_\delta}$ is the Gaussian Free Field in a non-trivial complex structure determined by the limit shape.

Consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$, with $d\in\{3,4,5\}$. Let $\{\varphi\geq0\}$ be its nonnegative excursion set, and consider the clusters obtained from…

Let $\theta\sim\mathbb{P}_{\beta,\mathbb{Z}^2}^{\operatorname{Vil}}$ be the Villain model at inverse temperature $\beta$, and let $\phi$ be a Gaussian free field with inverse tempe…

Wilson's conjecture. The scaling limits of these XOR-Ising loops can be described as the union of level sets of the Gaussian Free Field with an appropriately tuned spacing.

Consider the height functions associated with the $O(n)$ models, and let $\kappa$ and $\beta$ be the parameters describing their conjectured conformal scaling limits. Gaussian free…

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…

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…

Existence and uniqueness conjecture. In each polygon, there exists a unique global $\textnormal{-}\mathrm{SLE}_{}$ for each $$ -valenced link pattern $\in$ and for each $ \in (0,…

Gaussian free field scaling-limit conjecture. For a broad class of height functions, the scaling limit should be a continuum two-dimensional Gaussian free field.

Let $\Omega\subset\mathbb{R}^2$ be a bounded, simply connected domain, let $\alpha:\overline{\Omega}\to\mathbb{R}^2$ be a smooth vector field of the form $\alpha=\nabla V$, and let…

Let $alpha$ denote the parameter of the continuum fields $:!\cos(\alpha\phi)\!:$ and $:!\sin(\alpha\phi)\!:$, where $phi$ is the Gaussian free field. Nonexistence conjecture. No ex…

Let $\mathsf{Ht}_\mathcal N$ be the random height function of the model at scale $\mathcal N$, let $\mathbb E[\mathsf{Ht}_\mathcal N]$ be its expectation, let $\pounds$ denote the…

Let $\theta_6(N):=\mathbb{P}(\bm{0}\xleftrightarrow{\ge 0}\partial B(N))$ denote the critical one-arm probability for the metric graph Gaussian free field in dimension six, where…

Let $\widetilde{E}^{\ge 0}:=\{v\in\widetilde{\mathbb{Z}}^d:\widetilde{\phi}_v\ge 0\}$ be the nonnegative level set of the metric graph Gaussian free field, let…

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…

Let $\mathcal{K}^a$ denote the connected cluster associated with level $a$, let $\alpha$ and $\nu$ be the exponents in the paper, and let $\mathbb{P}$ denote the underlying Gaussia…

Critical-dimension volume conjecture. There exists a constant $\delta'>0$ such that

Critical-dimension one-arm conjecture. There exists a constant $\delta>0$ such that

Scaling-limit distinction conjecture. The scaling limit will be drastically different from a Brownian motion with time change, and in particular different from Liouville Brownian m…

Tilted SOS roughening conjecture. The slope should destabilize rigidity and induce Gaussian free field-type behavior of the surface at small $\beta$.

Let $\theta_6(N)$ denote the one-arm probability in the critical level set of the Gaussian free field on the metric graph of $\mathbb{Z}^6$. Critical one-arm order conjecture. Ther…

Let $d>6$, and consider sign clusters of the metric graph Gaussian free field, or equivalently loop clusters of the critical loop soup. Werner's conjecture. These clusters should b…