Void probabilities provide a fundamental description of point processes, expressing the probability that a given region contains no points. They fully characterize the law of any simple point process and appear in various guises across spatial statistics, stochastic geometry, and statistical physics. In Gibbs point processes, void probabilities are closely related, for example, to product densities, conditional intensities, the partition function, and consequently the likelihood function. Despite their theoretical importance, the void probabilities of Gibbs point processes have so far attracted limited attention as a topic of study in their own right.

In this talk, we develop a unified framework for studying void probabilities of Gibbs point processes and establish several new theoretical results. Central to our contribution is a cocycle relation for voids, which reveals a pseudo-factorization structure for void probabilities. This relation gives rise to a collection of analytical tools for exploring correlations between voids, as well as the influence of external configurations on local void probabilities. The resulting framework provides a versatile toolbox for studying, for example, correlation functions and constructing likelihood-based inference procedures in spatial statistics.