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 α-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 α-GAFs for any value of α. 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 α-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.

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强相关点场和广义高斯零系综中的近似吉布斯结构

随机点场中的吉布斯结构一直是研究其空间特性的经典工具。然而，精确的吉布斯性质仅在相对有限的模型类别中可用，并且它不能充分解决具有强依赖性空间结构的许多随机场。在这项工作中，我们为强相关随机点场（包括具有高度奇异空间结构的点场）的近似吉布斯结构提供了一个非常通用的框架。这些包括表现出很强的空间刚性的过程，特别是解析高斯零点场的某个单参数族，即α-GAF，已知它们表现出广泛的此类空间行为。我们的框架需要可以通过过程的有限粒子近似来验证的条件，我们将这种现象称为近似吉布斯属性。我们证明，这些使得人们能够将无限体积极限中的空间条件测量与适当奇异流形上支持的吉布斯型密度进行比较，我们将这种现象称为广义吉布斯性质。我们的工作提供了一种通用机制，可以严格理解随着系统规模不断增长的强相关点过程中空间调节的限制行为。我们通过证明对于任意 α 值，广义吉布斯性质对于 α-GAF 具有对数对势能，从而证明了我们方法的范围和多功能性。在这种情况下，我们肯定地解决了一个关于具有任何指定刚性水平的点过程是否存在的悬而未决的问题。特别是，对于 α-GAF 零过程，我们将刚性水平精确设置为$\lfloor \frac{1}{\alpha }\rfloor$，更不用说展示受局部保护影响的空间容忍现象$\lfloor \frac{1}{\alpha }\rfloor$时刻。对于涉及复杂的多体相互作用的此类过程，我们的结果表明随机点的局部行为仍然在短范围内表现出二维库仑型排斥。我们的技术可用于估计局部扰动下构型的相对能量，这可能对强相关随机点场上的动力学和随机几何产生影响。