Percolation models with long-range correlations via isomorphism theorems

渗流模型在统计力学中的应用已经有几十年的历史了。在20世纪40年代，化学诺贝尔奖获得者弗洛里和斯托克梅尔最初研究了它们在聚合物凝胶中的作用。从数学的角度来看，渗流理论的诞生是1957年布罗德本特和哈默斯利在煤矿工人防毒面具研究的推动下提出的伯努利渗流理论。该模型的主要特点之一是它的随机独立性，这简化了它的研究，并在这种背景下获得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的研究受到了越来越多的关注，其主要目的是利用同构定理对这类模型的两个典型例子--高斯自由场和随机交错模型的渗流特性进行精细分析。特别是，我们的目的是更好地了解高斯自由场的水平集以及随机交错的空集的渗流的关键参数。虽然这些方面的研究由于强相关性而本质上是困难的，但这些渗流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，如重整化群方法和解耦不等式，这些模型的性质可以通过复杂的同构定理被有利地转移到另一个模型中。我们计划获得的结果将提供对上述渗流模型中相变的某些方面的更深刻的理解，同时也回答了这一研究领域中的一些最重要的开放问题。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，我们期望它们将为马尔可夫环汤的渗流提供进一步的见解，最值得注意的是，也将对数论中的节点集的研究提供更多的见解。

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.