Conformal Maps and Planar Graphs

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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.