Cluster size distributions of discrete random fields

Abstract

We study size distributions of finite excursion clusters of discrete random fields on the integer lattice ${\mathbb{Z}}^d$. For stationary fields, we assign a unique root to each finite cluster and derive an exact representation of the root Palm size law through joint exceedance and exterior nonexceedance probabilities for complete connected shapes. Root sampling gives each finite cluster equal weight and remains well defined even when infinite components are present. We also study the cluster size law conditional on a specified site being a local maximum above the threshold $u$, providing a local formulation for nonstationary fields. Under stationarity and almost sure cluster finiteness, an exact change of measure shows that peak sampling reweights the root law by the number of local maxima in each cluster. For stationary Gaussian fields satisfying suitable conditions, the total variation distance between these laws decreases exponentially in $u^2$. Guarded spatial Monte Carlo yields unbiased estimators of cluster intensities, estimates of sampling uncertainty from independent realizations, and a finite sample certificate for root truncation error. We further establish sufficient exponential cluster size tail bounds at a fixed threshold and exact logarithmic Gaussian decay rates at a fixed cluster size as $u\to \infty$. The latter rates are characterized by quadratic programs retaining the exterior constraints needed to identify complete components. Numerical illustrations compare direct probability evaluations with Monte Carlo estimates and assess sampling uncertainty and truncation error. The general representations and estimators apply to both Gaussian and non-Gaussian fields, providing a probabilistic foundation for quantifying spatial extent in discretely sampled data.