Percolation models with long-range correlations via isomorphism theorems

几十年来，渗流模型一直在统计力学中扮演着重要的角色。20世纪40年代，诺贝尔化学奖得主弗洛里和斯托克梅耶首先在聚合物的凝胶化过程中对它们进行了研究。从数学的角度来看，渗透理论的诞生是1957年Broadbent和Hammersley在对煤矿工人防毒面具的研究的推动下引入伯努利渗透理论。该模型的一个重要特征是它的随机无关性，这简化了它的研究，并得到了非常深刻的数学结果。近年来，对具有强相关性的渗流模型更真实、更复杂情况的研究越来越受到人们的重视。该项目的主要目标是通过使用同构定理，对这种模型的两个标志性例子，高斯自由场和随机交错模型的具体渗透方面进行精细分析。特别是，我们的目标是更好地理解高斯自由场的水平集和随机交错的空集的渗透的关键参数。虽然由于强相关性，这些方面的研究本质上是困难的，但这些渗透模型的同构定理等工具的最新发展开辟了新的视角：结合强大而成熟的技术，如重整化群方法以及解耦不等式，可以通过复杂的同构定理的使用来有益地转移其中一个模型的属性，从而获得对另一个模型的有趣见解。我们计划获得的结果将对上述渗流模型中相变的某些方面提供更深刻的理解，同时回答该研究领域中一些最重要的开放性问题。更重要的是，该项目开发的工具也有望对概率论和数学的其他领域产生影响。特别是，我们期望他们将进一步深入了解马尔可夫环汤的渗透，最值得注意的是，也将深入研究数论中的节点集。

Percolation models have been playing a fundamental role in statistical mechanics for several decades by now. They have initially been investigated in the gelation of polymers during the 1940s by chemistry Nobel laureate Flory and then Stockmayer. From a mathematical point of view, the birth of percolation theory was the introduction of Bernoulli percolation by Broadbent and Hammersley in 1957, motivated by research on gas masks for coal miners. One of the key features of this model is its stochastic independence which simplifies its investigation, and very deep mathematical results have been obtained in this setting. During recent years, the investigation of the more realistic and at the same time more complex situation of percolation models with strong correlations has attracted more and more attention.The main goal of this project is the refined analysis of specific percolative aspects of two emblematic examples of such models, the Gaussian Free Field and the model of Random Interlacements, via the use of isomorphism theorems. In particular, we aim at obtaining a better understanding of the critical parameters for the percolation of level sets of the Gaussian Free Field as well as of the vacant set of Random Interlacements. While the investigation of such aspects is intrinsically difficult due to the strong correlations, the recent development of tools such as isomorphism theorems for these percolation models opens up novel perspectives: In combination with powerful and established techniques such as renormalization group methods as well as decoupling inequalities, properties of one of these models can be beneficially transferred via the sophisticated use of isomorphism theorems to derive interesting insights into the other model.The results we plan to obtain will provide a more profound understanding of certain aspects of the phase transition in the above percolation models, and at the same time answer some of the most important open problems in this field of research. What is more, the tools developed along the project are also expected to have implications on other fields of probability theory and mathematics. In particular, we expect that they will provide further insights into the percolation of Markov loop soups and, most notably, into the investigation of nodal sets in number theory also.