Structural properties of Gibbs process mixtures

This thesis studies the mixture and disintegration structure of Gibbs point processes. Motivated by the question of whether Gibbsian properties are preserved under randomization, we develop a rigorous framework for mixtures of Gibbs measures and mixtures of Papangelou interaction structures. Rather than starting from a prescribed interaction and constructing a probability law, we begin with families of probability laws and ask how their mixtures admit a Gibbsian description. Under suitable measurability and kernel assumptions, the resulting mixture law remains a Gibbs point process. Its Papangelou intensity is obtained by averaging jointly measurable versions of the component intensities against the posterior distribution of the mixing parameter given the configuration. The proof uses disintegration and the Georgii–Nguyen–Zessin identity. We also study the passage from a measurable family of Papangelou intensities to a probability kernel of Gibbs measures under component uniqueness assumptions. The developed theory is applied to Gibbsian cluster point processes and parametric hard-core point processes. For cluster models, the parent configuration is treated as the mixing object, leading to a Gibbsian representation of the resulting cluster process under the stated assumptions. For hard-core models, the hard-core radius is treated as a random parameter. In the infinite-volume model considered here, under the assumed uniqueness regime and observation of the entire configuration, the posterior distribution of this parameter is concentrated at the infimum of interpoint distances. Finally, we formulate a uniqueness question within the class of probability laws mutually absolutely continuous with a given Gibbs law. A Poisson mixture example shows that uniqueness of the component Gibbs measures does not imply uniqueness of the mixture law for a common Papangelou intensity.