We begin by reviewing one-dimensional random objects that are universal in the sense that they arise in many contexts—in particular as scaling limits of large families of discrete models—and canonical in the sense that they are uniquely characterized by scale invariance and other natural symmetries. Examples include Brownian motion, Bessel processes, stable L’evy processes and ranges of stable subordinators.
We then introduce several universal and canonical random objects that are (at least in some sense) two dimensional or planar, along with discrete analogs of these objects. These include trees, distributions, curves, loop ensembles, surfaces, and growth trajectories. Keywords include continuum random tree, stable L’evy tree, stable looptree, Gaussian free ﬁeld, Schramm-Loewner evolution, percolation, uniform spanning tree, loop-erased random walk, Ising model, FK cluster model, conformal loop ensemble, Brownian loop soup, random planar map, Liouville quantum gravity, Brownian map, Brownian snake, diffusion limited aggregation, ﬁrst passage percolation, and dielectric breakdown model.
Finally, we discuss the intricate and surprising relationships between these universal objects. We explain how to use generalized functions to construct curves and vice versa; conformally weld a pair of surfaces to produce a surface decorated by a simple curve; topologically and conformally mate pairs of trees to obtain surfaces decorated by non-simple curves; and reshuffle these constructions to describe random growth trajectories on random surfaces. We present both discrete and continuum analogs of these relationships. Keywords include imaginary geometry, quantum zipper, peanosphere, and quantum Loewner evolution.
The lecture notes are a draft in progress by Scott Sheffield and Jason Miller.
Lecture Notes for Universal Random Structures in 2D (PDF) (Courtesy of Scott Sheffield and Jason Miller. Used with permission.)