Conformal Maps and Planar Graphs

在统计物理的许多模型中，例如铁磁性的伊辛模型，图案和形状都是从随机性中产生的。类似的形状可以在各种现实生活中观察到，因为这些模型旨在描述自然现象。自相似集合（如雪花）在不同的尺度上看起来完全相同，而PI研究的随机集合在不同的尺度上看起来只是大致相同，它们的外观由某种概率定律描述。虽然这些集合很容易观察，也不难模拟，但它们很难用数学方法来处理。在过去的十年里，人们对这种随机形状的理解取得了巨大的进步，但仍有许多基本问题有待解决。PI的工作旨在解决其中的一些问题，并试图提供工具来研究随机集的形状。上述进展主要基于尺度极限的保形不变性，并将其与Schramm-Loewner演化SLE联系起来。例如，根据Smirnov等人的工作，临界时Ising模型的界面是SLE(3)的形式，外环集合的标度极限是保形环系CLE(3)。PI使用来自圆填充的工具研究CLE的共形表示，旨在模拟Bonk的（确定性）Sierpinski地毯的均匀化定理。PI还研究了平面图形的共形表示，使用具有锥奇点的平面的均匀化，以及它们的Shabat多项式或Belyi函数的表示。一个主要的目标是建立一个存在的共形自然嵌入随机树的分布极限，当边的数量趋于无穷。一个关键的工具是光盘层合的保形焊接。

In many models of statistical physics, such as the Ising model for ferromagnetism, patterns and shapes emerge from randomness. Similar shapes can be observed in various real-life situations, as the models aim to describe natural phenomena. Whereas self-similar sets such as snowflakes look exactly the same at different scales, the random sets the PI studies look only roughly the same at different scales, with their appearance being described by some probabilistic law. While these sets are easy to observe and not hard to simulate, they are difficult to treat mathematically. The last decade has seen dramatic progress in the understanding of such random shapes, but many fundamental questions are still open. The PI's work aims towards resolutions of some of these questions, and tries to provide tools to investigate the shapes of random sets.The aforementioned progress is largely based on conformal invariance properties of the scaling limits, connecting it to the Schramm-Loewner evolution SLE. For instance, by work of Smirnov and others, interfaces of the Ising model at criticality are forms of SLE(3), and the scaling limit of the collection of outer loops is the Conformal Loop Ensemble CLE(3). The PI investigates conformal representations of CLE's using tools from circles packings, aiming at an analog of Bonk's uniformization theorem for(deterministic) Sierpinski carpets. The PI also studies conformal representations of planar graphs using uniformizations of flat surfaces with cone singularities, as well as their representations as Shabat polynomials or Belyi functions. A main goal is to establish the existence of a distributional limit of the conformally natural embedding of random trees as the number of edges tends to infinity. A key tool is the conformal welding of laminations of the disc.