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.

在我的博士研究中，我建立了两个著名的欧几里得场论的全局极大值的收敛结果，sin - gordon场和d=2的Phi-4场。为此，我开发了一套工具，将感兴趣的领域与研究得很好的高斯自由场结合起来。特别是，这种技术允许比较两个字段的极值。我的技术适用于更多的例子，我的目标是为它们的极端值建立一个统一的理论。此外，我将重点研究相关的统计领域，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的结果。在这里，我的目标是调整重整化的论点，以适应这些不同的背景，并回答这一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在phil -4领域的一个项目中建立的。

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.