Percolation models with long-range correlations via isomorphism theorems

点击翻译按钮获取中文摘要

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.