Near-critical two-dimensional random systems

这个项目研究随机空间系统，如渗透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：这个参数的特定临界值有一个急剧的转变。在临界点上，随机宏观几何——通常是分形几何——出现，在二维中，这些几何被广泛认为具有强的保形不变性。在Lawler， Schramm， Smirnov和Werner的突破之后，对这些模型的数学理解在过去十年中有了很大的改善，特别是由于平面随机增长的共形不变Schramm- loewner - evolution过程。该项目解决了与这些模型在临界和接近临界时的行为有关的问题，以及相关的“自临界”系统-例如森林火灾模型-在没有任何参数微调的情况下本质上出现相变。这里研究的概率论领域与组合学和复分析重叠，它也使用了统计力学的思想和技术。随机形状，如由两种金属焊接而成的粗糙界面，或由侵蚀形成的不规则海岸，在自然界中无处不在。这些形状通常表现为分形行为，概率论中越来越重要的一部分致力于研究这种空间随机性起中心作用的模型。这导致了深刻而迷人的数学问题，特别是令人惊讶的“普遍性”特性出现：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程序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.