Dr Laurie Field, Australian National University & Dr Gregory Markowsky, Monash University
One of the most exciting recent developments in probability theory and mathematical physics has been the discovery of so-called “conformal invariance” in two-dimensional models. Physicists have long conjectured, extrapolating from symmetries under rotation and dilation, that many of their discrete models in the plane would have conformally invariant scaling limits — that is, processes which have the same law after the underlying geometry is mapped in a continuous, angle-preserving way. Recent work by mathematicians has shown this to be correct, as well as identifying the natural limiting process of interfaces in many of these models as the Schramm–Loewner Evolution (SLE). This theory has led to the rigorous determination of critical exponents such as the Brownian intersection exponents and the Hausdorff dimensions of many random planar fractals.
In the first half of this course, we study the most important discrete models that exhibit conformal invariance in the scaling limit, including simple random walk, loop-erased walk, percolation and the Ising model, and discuss what forms of discrete complex analysis can be used to illuminate these models.
In the second half of the course, we pass to the continuum, where Brownian motion becomes a key tool. Lévy’s theorem on conformal invariance of planar Brownian motion yields quick and informative proofs of many facts from complex analysis. Loewner’s theorem in complex analysis describes a non-crossing curve by the differential equation satisfied by the uniformizing conformal map from its complement. By applying Loewner’s differential equation to a Brownian motion on the boundary, we obtain the definition of SLE, derive its first important properties, and heuristically explain why it is the scaling limit of interfaces in the models studied in the first half.
- Simple random walk and its convergence to Brownian motion;
- Loop-erased random walk and the uniform spanning tree;
- Critical percolation and Smirnov’s proof of conformal invariance;
- The Ising model and its discrete holomorphic fermions;
- Self-avoiding walk and the connective constant on the hexagonal lattice;
- Lévy’s Theorem on the conformal invariance of Brownian motion;
- Analytic functions and conformal maps, including Möbius transformations and other elementary maps, as well as the behaviour of Brownian motion under these transformations;
- The Poisson Integral Formula and its relation to Brownian motion;
- (Non-random) Loewner Evolution, which is the conformal theory associated with curves evolving in the plane;
- We will conclude with the basics of the Schramm–Loewner Evolution and its relation to the discrete physical processes discussed in the first half of the course.
The required background for this course is a basic understanding of probability theory and analysis, including measure theory. Knowledge of complex analysis would be an advantage, but the relevant concepts (discrete and continuous) will be introduced during the course.
- Mid-School assignment: 40%
- Final examination: 60%
The primary resource will be lecture notes which will be provided by the lecturers. The following texts are good references for the topics to be covered.
- “Probability on Trees and Networks” by R. Lyons and Y. Peres.
- “Percolation and the Ising Model” by W. Werner.
- “Brownian Motion and Martingales in Analysis” by R. Durrett.
Dr Gregory Markowsky, Monash University
Greg is a Senior Lecturer at Monash University. Before that, he was a postdoctoral researcher at Pohang University of Science and Technology in Korea and worked for Wolfram Research on the WolframAlpha website. His PhD was obtained at City University of New York under Jay Rosen. His research has focused on various aspects of probability theory, including connections between Brownian motion and complex analysis, graph theory, Markov chains, and Gaussian processes. More information can be found on his webpage: http://users.monash.edu.au/~gmarkow/
Dr Laurie Field, The Australian National University
Laurie is a Lecturer at the ANU. Prior to that, he was a postdoctoral research fellow at EPFL in Switzerland, after completing a PhD under Greg Lawler at the University of Chicago. Laurie’s research focuses on conformally invariant objects in probability theory, particularly Brownian motion, loop measures and SLE, and their connections to mathematical physics. More information is available on his webpage: http://sma.epfl.ch/~field/