Percolation models with long-range correlations via isomorphism theorems

逾渗模型在统计力学中发挥着重要的作用已经有几十年了。它们最初是在20世纪40年代由诺贝尔化学奖获得者Flory和Stockmayer在聚合物的凝胶化中进行研究的。从数学的角度来看，渗流理论的诞生是由布罗德本特和哈默斯利在1957年引入伯努利渗流，其动机是对煤矿工人防毒面具的研究。该模型的一个关键特征是它的随机独立性，这简化了其调查，并在此设置中获得了非常深刻的数学结果。近年来，对强相关渗流模型的研究越来越受到人们的关注，本课题的主要目标是利用同构定理对这类模型的两个典型例子--高斯自由场模型和随机交错模型的具体性质进行精细分析。特别是，我们的目标是获得一个更好的理解的临界参数的渗透水平集的高斯自由场以及空的随机交错集。虽然由于强相关性，这些方面的调查本质上是困难的，但最近开发的工具，如这些渗流模型的同构定理，开辟了新的视角：结合强大的和既定的技术，如重整化群方法以及解耦不等式，这些模型中的一个的性质可以通过复杂的同构定理的使用来有益地转移到另一个模型中，以获得有趣的见解。我们计划将结果获得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，同时回答该研究领域中一些最重要的开放问题。更重要的是，沿着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，我们希望他们将提供进一步的见解马尔可夫回路汤的渗流，最值得注意的是，在数论的节点集的调查也。

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.