5 problems

Let $\Lambda_n$ be the finite domain and let $\psi_n$ be a Villain model in $\Lambda_n$ at sufficiently small temperature $T$. Let the discrete ALE be the union of the interfaces a…

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 $\beta$ be the inverse temperature of the Villain model, and let the natural interface be the interface constructed from the Villain spins in the scaling limit. Villain-interfa…

Let $\beta>0$ be the inverse temperature, let \boldsymbol{\theta}=(\theta_x)_{x\in\\{-n,\ldots,n\}^2} denote Villain spins, and let $\beta'$ satisfy…