2 problems

Gaussian fluctuation criterion. Under these assumptions, the convergence above should hold if and only if the shallow set $S^*\setminus S$ has capacity zero.

Let $Q$ be the planar potential defining the exponentially varying measure, let $\mathcal{S}_1$ be its droplet, and let $z_0\in\partial\mathcal{S}_1$ be a point where the boundary…