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

Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated random point fields, including those with a highly singular spatial structure. These include processes that exhibit strong spatial rigidity, in particular, a certain one-parameter family of analytic Gaussian zero point fields, namely the $\alpha$-GAFs, that are known to demonstrate a wide range of such spatial behavior. Our framework entails conditions that may be verified via finite particle approximations to the process, a phenomenon that we call an approximate Gibbs property. We show that these enable one to compare the spatial conditional measures in the infinite volume limit with Gibbs-type densities supported on appropriate singular manifolds, a phenomenon we refer to as a generalized Gibbs property. Our work provides a general mechanism to rigorously understand the limiting behavior of spatial conditioning in strongly correlated point processes with growing system size. We demonstrate the scope and versatility of our approach by showing that a generalized Gibbs property holds with a logarithmic pair potential for the $\alpha$-GAFs for any value of $\alpha$. In this vein, we settle in the affirmative an open question regarding the existence of point processes with any specified level of rigidity. In particular, for the $\alpha$-GAF zero process, we establish the level of rigidity to be exactly $\lfloor \frac{1}{\alpha }\rfloor$, a fortiori demonstrating the phenomenon of spatial tolerance subject to the local conservation of $\lfloor \frac{1}{\alpha }\rfloor$ moments. For such processes involving complex, many-body interactions, our results imply that the local behavior of the random points still exhibits 2D Coulomb-type repulsion in the short range. Our techniques can be leveraged to estimate the relative energies of configurations under local perturbations, with possible implications for dynamics and stochastic geometry on strongly correlated random point fields.