4 problems

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 ……

Exponential-profile conjecture. The limit

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…