Cluster size distributions of discrete random fields

Abstract: We study discrete random fields ${X_t: t\in \mathbb{Z}<sup>d}$ parameterized on the $d$-dimensional integer lattice $\mathbb{Z}<sup>d$. For a fixed threshold $u$, the excursion set ${t \in \mathbb{Z}^d : X_t &gt; u}$ decomposes into connected components or clusters, whose size, defined as the number of lattice points they contain, are random. This paper investigates the probability distribution of these cluster sizes. For stationary random fields, we derive exact expressions for the cluster size distribution. To address nonstationary settings, we introduce a peak-based cluster size distribution, which characterizes the distribution of cluster sizes conditional on the presence of a local maximum above $u$. This formulation provides a tractable alternative when exact cluster size distributions are analytically inaccessible. The proposed framework applies broadly to Gaussian and non-Gaussian random fields, relying only on their joint dependence structure. Our results provide a theoretical foundation for quantifying spatial extent in discretely sampled data, with applications to medical imaging, geoscience, environmental monitoring, and other scientific areas where thresholded random fields naturally arise.