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Gibbsian structure in random point fields has been a classical tool forstudying their spatial properties. However, exact Gibbs property is availableonly in a relatively limited class of models, and it does not adequatelyaddress many random fields with a strongly dependent spatial structure. In thiswork, we provide a general framework for approximate Gibbsian structure forstrongly correlated random point fields. These include processes that exhibitstrong spatial rigidity, in particular, a certain one-parameter family ofanalytic Gaussian zero point fields, namely the $\alpha$-GAFs. Our frameworkentails conditions that may be verified via finite particle approximations tothe process, a phenomenon that we call an approximate Gibbs property. We showthat these enable one to compare the spatial conditional measures in theinfinite volume limit with Gibbs-type densities supported on appropriatesingular manifolds, a phenomenon we refer to as a generalized Gibbs property.We demonstrate the scope of our approach by showing that a generalized Gibbsproperty holds with a logarithmic pair potential for the $\alpha$-GAFs for anyvalue of $\alpha$. This establishes the level of rigidity of the $\alpha$-GAFzero process to be exactly $\lfloor \frac{1}{\alpha} \rfloor$, settling in theaffirmative an open question regarding the existence of point processes withany specified level of rigidity. For processes such as the zeros of$\alpha$-GAFs, which involve complex, many-body interactions, our results implythat the local behaviour of the random points still exhibits 2D Coulomb-typerepulsion in the short range. Our techniques can be leveraged to estimate therelative energies of configurations under local perturbations, with possibleimplications for dynamics and stochastic geometry on strongly correlated randompoint fields.

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles