5 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.

Gaussian-fluctuation conjecture. The limit defining $q_{c,\alpha}$ exists and satisfies the stated asymptotic decoupling of occupation probabilities. Furthermore,…

Let $D$ be the domain, let $\theta^{\varepsilon}$ have the Gibbs measure with external field $h>0$ given by, and let $\eta^{\varepsilon}$ be the associated gradient field defined i…

Let $\psi_j^0$ and $\psi_i^t$ denote eigenvectors at times $0$ and $t$, respectively. Fix an index $j=j_N$ such that $j_N/N\to y$, and let $i_1<\dots<i_n\in\{1,\dots,N\}$ satisfy…

Let $N$ denote the size parameter of the particle system, and consider the height function at a single point. One-point fluctuation conjecture. After scaling by $\sqrt{\log N}$, th…