Near-critical two-dimensional random systems

这个项目研究随机空间系统，如渗流或铁磁性的伊辛模型。这些模型都表现出相变的统计物理现象，它们的宏观行为随着一个参数的变化而急剧变化：该参数的一个特定临界值存在一个急剧的转变。正是在临界点，随机的宏观几何--通常是分形的--几何产生，并且在二维中这些几何被广泛认为具有很强的共形不变性。随着Lawler，Schramm，Smirnov和Werner的突破，特别是由于平面上随机增长的共形不变Schramm-Loewner演化过程，对这些模型的数学理解在过去十年中有了很大的提高。这个项目解决了与这些模型在临界点和临界点附近的行为有关的问题，以及相关的“自临界”系统--例如森林火灾模型--在这种系统中，相变复杂地出现，而没有对参数进行任何微调。这里研究的概率论领域交叠了组合学和复数分析，还运用了统计力学的思想和技术。随机形状，如两个金属焊接形成的粗糙界面，或由侵蚀形成的不规则海岸，在自然界中无处不在。这些形状通常表现出一种分形行为，概率论的一个越来越重要的部分致力于研究这种模型，其中空间随机性起着核心作用。这导致了深刻而有趣的数学问题，特别是令人惊讶的“普遍性”性质的出现：例如，相似的形状出现在乍一看完全不相关的情况下，基于完全不同的物理、化学或生物机制。对简化模型的更好的数学理解将为应用中使用的更复杂的模型提供新的见解。

This project studies random spatial systems, such as percolation or the Ising model of ferromagnetism. These models all exhibit the statistical physics phenomenon of a phase transition, their macroscopic behavior changing drastically as one parameter varies: there is a sharp transition for a particular critical value of this parameter. At the critical point exactly, random macroscopic - most often fractal - geometries arise, and in two dimensions these geometries are widely believed to possess a strong property of conformal invariance. The mathematical understanding of these models has considerably improved over the last ten years following breakthroughs of Lawler, Schramm, Smirnov and Werner, in particular thanks to the conformally invariant Schramm-Loewner-Evolution process of stochastic growth in the plane. This project addresses questions related to the behavior of these models at and near criticality, as well as related "self-critical" systems - forest-fire models for instance - where a phase transition intrisically appears without any fine-tuning of a parameter. The area of probability theory studied here overlaps combinatorics and complex analysis, and it also uses ideas and techniques from statistical mechanics.Random shapes, such as rough interfaces created by welding two metals, or irregular sea coasts fashioned by erosion, are omnipresent in nature. These shapes usually display a fractal behavior, and an increasingly important part of probability theory is devoted to studying such models where spatial randomness plays a central role. This leads to deep and fascinating mathematical questions, in particular surprising "universality" properties arise: for instance, similar shapes appear in situations that are, at first sight, completely unrelated, based on totally different physical, chemical or biological mechanisms. A better mathematical understanding of simplified models would provide new insight on more complex models used in applications.