12 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…

Let $d=2$ and suppose that Assumption-(i) and the moment condition hold, in particular $\operatorname{\mathbb{P}}[\omega(e)>0]=1$ for $e\in E_d$. For $n\in\mathbb{N}$, set … and ……

Let $h_*$ and $u_*$ denote, respectively, the critical parameters for level-set percolation of the discrete Gaussian free field and for percolation of the vacant set of random inte…

Let $D\subset\mathbb{R}^2$ be a bounded domain, and let $\mu$ be the random measure appearing in the limiting extremal process of the discrete Gaussian free field, characterised by…

Let $Z(dx)$ be the random measure appearing in the limiting extremal process of the two-dimensional discrete Gaussian free field (DGFF), whose intensity is $e^{-2x}\otimes Z(dx)$.…

Near-extremal-profile conjecture. There is a constant $c>0$ such that

Exponential-profile conjecture. The limit

Deep-extremal-level-set size conjecture. There is $c\in(0,\infty)$ such that

Let $\mathfrak D$ be the class of domains in the source, let $D_N$ be the corresponding lattice approximation, let $h_z$ be the discrete Gaussian free field, let $m_N$ be its cente…

Let $D$ be a domain, let $t\in\mathbb R$, and let $\rho^D(t,\mathord\cdot)$ denote the density from Theorem 2.7. Boundary regularity conjecture. For each $t\in\mathbb R$, the funct…

Let $\mathfrak D$ be the class of domains under consideration, and let $\{M^D\colon D\in\mathfrak D\}$ be a family of random Borel measures satisfying conditions (0)--(3) of Theore…

The random Borel measure $Z^D$ is defined on a domain $D\in\mathfrak D$ as the limiting extremal measure associated with the two-dimensional discrete Gaussian free field. Critical…