Analysis of the extreme values for certain non-Gaussian and Gaussian fields, in particular the Ginzburg-Landau gradient interface model in d=2 and log-correlated Gaussian fields on percolation clusters using methods from renormalisation group theory.

使用重正化群理论的方法分析某些非高斯和高斯场的极值，特别是 d=2 中的 Ginzburg-Landau 梯度界面模型和渗滤簇上的对数相关高斯场。

• 批准号: 523936652
• 负责人: Michael Hofstetter
• 金额: --
• 依托单位: Computer Science & Applied Mathematics
• 依托单位国家: 德国
• 项目类别: WBP Fellowship
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/523936652?language=en
• 关键词: Analysis; extreme; values; certain; non

In my doctoral studies, I established convergence results for the global maximum of two distinguished Euclidean field theories, the sine-Gordon field and the Phi-4 field in d=2. To this end, I developed a set of tools to couple the field of interest with the well-studied Gaussian free field. In particular, this technique allows to compare the extreme values of both fields. There are many more examples to which my techniques apply, and it is my goal to establish a unified theory for their extreme values. Moreover, I will focus on related statistical fields, such as the Ginzburg-Landau gradient interface model and the Gaussian free field on disordered graphs, for which initial results already exists thanks to the work of my postdoctoral supervisor Ofer Zeitouni. Here, my goal is to adjust the renormalisation arguments to these different settings and answer some of the many open questions in this area. A crucial result that I hope to further exploit is the equivalence between the Polchinski renormalisation group approach and the Boue-Dupuis stochastic control representation, which I established in a recent project with N. Barashkov and T. Gunaratnam on the Phi-4 field.

在我的博士研究中，我建立了两个著名欧几里得场论（即 d=2 时的正弦戈登场和 Phi-4 场）的全局最大值的收敛结果。为此，我开发了一套工具将感兴趣的场与经过充分研究的高斯自由场耦合起来。特别是，该技术允许比较两个字段的极值。我的技术适用的例子还有很多，我的目标是为它们的极端值建立一个统一的理论。此外，我将重点关注相关的统计领域，例如Ginzburg-Landau梯度界面模型和无序图上的高斯自由场，由于我的博士后导师Ofer Zeitouni的工作，这些领域已经有了初步结果。在这里，我的目标是根据这些不同的设置调整重整化参数，并回答该领域的一些悬而未决的问题。我希望进一步利用的一个关键结果是 Polchinski 重正化群方法与 Boue-Dupuis 随机控制表示之间的等价性，这是我在最近与 N. Barashkov 和 T. Gunaratnam 合作的 Phi-4 域项目中建立的。