Lectures: Tuesday and Thursday 1:00-2:30, E17-122. First lecture Thursday, September 10.
Office hours: Tuesday 2:30-3:30 and Thursday 2:30-3:30 in E17-312.
Assignments: three problem sets and one final project. Final project may be either expository or original-research based. Several suggested research problems will be presented. Collaborative efforts will be allowed.
Texts: course notes and references to be assigned.
Prerequisites: basic probability at the level of an introductory graduate course (18.175 or equivalent).
Textbook: Lecture notes with Jason Miller in progress
Course topics: first a brief review of universal random structures in 1D, including Brownian motion, Bessel processes, stable Levy processes, ranges of stable subordinators, and Ito's formula. Then an introduction to universal random structures that are (at least in some sense) two dimensional or planar, including planar trees, generalized functions on planar domains, Riemannian surfaces, planar growth models, planar loop ensembles, and planar connections. Discussion of motivating problems from statistical physics, quantum field theory, conformal field theory, string theory, and early universe cosmology.
1. Random planar trees: Aldous's continuum random tree, Levy trees, loop trees, Brownian snakes, Galton Watson trees and uniform random trees.
2. Random generalized functions: Gaussian free fields (free boundary, fixed boundary, massive), fractional Gaussian fields, log correlated free fields, discrete Gaussian free field, dimer model height functions, uniform spanning tree height functions, non-intersecting lattice paths and determinants, Laplacian determinants.
3. Random curves: Schramm-Loewner evolution (SLE), conformal loop ensembles (CLE), 2D Brownian motion, percolation, Ising and Potts models, FK cluster models, GFF level lines/harmonic explorer, uniform spanning tree boundary, loop-erased random walk, Wilson's algorithm.
4. Random surfaces: Brownian map and Liouville quantum gravity, multiplicative chaos, random planar maps, random quadrangulations, random triangulations, the Schaeffer bijection, the Mullin bijection, the FK bijection.
5. Random growth models: KPZ growth, Brownian web, Hastings-Levitov, DLA, Eden model, internal DLA.
6. Random connections: Yang Mills, quantum electromagnetism, lattice Yang Mills.
1. Imaginary geometry: generalized functions and curves.
2. Conformal welding: surfaces, generalized functions and curves.
3. Mating trees and the peanosphere: trees, surfaces, generalized functions and curves.
4. Quantum Loewner evolution and the Brownian map: growth models, trees, surfaces, generalized functions and curves.
Graduate probabiliy background
Continuum random tree
Random planar maps and the Brownian map
Gaussian free field
Liouville quantum gravity
Schramm-Loewner evolution and discrete analogs
GFF + SLE
LQG + LQG = LQG + SLE
CRT + CRT = LQG + SLE
LQG + reshuffled SLE = LQG + DBM
LQG = TBM
Open problem document in progress
Problem set document