Cluster size distributions of discrete random fields

We study discrete random fields $X_t$ indexed by the integer lattice $\mathbb{Z}^d$. For a fixed threshold $u$, the excursion set $\{t \in \mathbb{Z}^d : X_t > u\}$ decomposes into connected components, called excursion clusters, whose sizes are the numbers of lattice sites they contain. Our objecti