Delocalization conjecture for two-dimensional integer-valued random surfaces with general potentials

Let $U:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $U(x)=U(-x)$ and

$$\lim_{x\to\infty}\frac{U(x)}{x^{\alpha}}=\infty$$

for some $\alpha>0$. For each $M>0$, define the interaction sequence $u^M=(u^M_n)$ by $u^M_n:=U(n/M)$. For the two-dimensional box $\Lambda_L$ with free boundary conditions, let $\mathbb{E}_{u^M,\Lambda_L^{\operatorname{free}},(0,0)}^{\operatorname{IV}}$ denote expectation for the corresponding integer-valued random surface pinned at $(0,0)$. Delocalization conjecture. There exists $M_0(U)>0$ such that, for every $M>M_0(U)$,

$$\lim_{L\to\infty}\frac{\mathbb{E}_{u^M,\Lambda_L^{\operatorname{free}},(0,0)}^{\operatorname{IV}}\left(m_{(L-1,L-1)}^2\right)}{\log(L)}>0.$$

In the special case $U(x)=|x|^\alpha$, this predicts delocalization for interactions $u_n=\beta|n|^\alpha$ in two dimensions at sufficiently low $\beta$, extending the Fröhlich–Spencer results for $\alpha=1,2$; the conjecture remains open for the stated general class of potentials.