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 general framework for approximate Gibbsian structure for
    strongly correlated random point fields. 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. 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.
    We demonstrate the scope of our approach by showing that a generalized Gibbs
    property holds with a logarithmic pair potential for the $\alpha$-GAFs for any
    value of $\alpha$. This establishes the level of rigidity of the $\alpha$-GAF
    zero process to be exactly $\lfloor \frac{1}{\alpha} \rfloor$, settling in the
    affirmative an open question regarding the existence of point processes with
    any specified level of rigidity. For processes such as the zeros of
    $\alpha$-GAFs, which involve complex, many-body interactions, our results imply
    that the local behaviour 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.

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