Universal Randomness in Dimension 2

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This research is in probability theory. It will attempt to make connections between a number of different probabilistic models that arise in statistical physics and particle physics. Understanding these connections will result in improved understanding of the physical models. The PI will train students from the undergraduate level through the graduate level. He will also assist in the training of postdoctoral researchers. This research concerns a variety of random fractals that are in some sense planar (i.e., naturally embedded in or parameterized by subsets of the plane). These fractals include random paths, random surfaces, random collections of loops, random trees, random functions, and random growth processes. The objects under study are "universal" in the sense that they arise as limits of many discrete models and "canonical" in the sense that they are uniquely characterized by special symmetries. Several of these objects are motivated by statistical mechanics, string theory, and gauge theory, as well as the study of natural growth processes (lichen, mineral depositions, snowflakes, lightning bolts, etc.) The specific technical goals of the research include the following: understanding the limiting conformal structure of discrete planar maps, generalizing quantum Loewner evolution growth models beyond the regime in which they have been defined, endowing general Liouville quantum gravity surfaces with metric structure, extending known results about random surface scaling limits to random surfaces embedded in higher dimensional spaces, and understanding more about the relationship between lattice Yang-Mills gauge theory (and its variants) and the random surface models that arise in the formulas for Wilson loop expectations. The broader aim is to provide a firmer mathematical understanding of some of our most fundamental models for physical phenomena, ranging from microscopic to macroscopic scales.