18.177 Universal random structures in 2D: Fall, 2015

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.

Summary of universal objects and discrete analogs:

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.

Relationships among universal objects:

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.

Selected references on universal objects


  • Introductory slides

    Graduate probabiliy background

  • Probability: theory and examples (Durrett)
  • Slides and other references from 18.175

    Continuum random tree

  • Random trees, Levy processes, and spatial branching processes (Duquesne and Le Gall)
  • Levy processes, stable processes, and subordinators (Lalley)
  • Bessel processes (Lawler)
  • Continuous martingales and Brownian motion (Revuz and Yor)
  • Levy processes (Bertoin)
  • Poisson point proccesses (Johnson)
  • Stable loop trees (Curien and Kortchemski)
  • Continuum random tree I, Continuum random tree II, Continuum random tree III (Aldous)
  • Brownian motion (Morters and Peres)
  • Moore's theorem (Timorin)
  • Random planar maps and the Brownian map

  • Scaling limits of random trees and planar maps (Le Gall and Miermont)
  • Random geometry on the sphere (Le Gall)
  • Slides on Cori-Vauquelin-Schaeffer bijection and Brownian map convergence (Bernardi)
  • Quantum gravity and inventory accumulation (Sheffield)
  • An axiomatic characterization of the Brownian map (Miller, Sheffield)
  • Gaussian free field

  • Gaussian free fields for mathematicians (Sheffield)
  • Topics on the two-dimensional Gaussian free field (Werner)
  • Log-correlated free field in general dimension (Duplantier, Rhodes, Sheffield, Vargas)
  • Fractional Gaussian fields: a survey (Lodhia, Sheffield, Sun, Watson)
  • Liouville quantum gravity

  • Liouville Quantum Gravity and KPZ (Duplantier and Sheffield)
  • Quantum gravity and the KPZ formula (Garban)
  • Introduction to the Gaussian Free Field and Liouville Quantum Gravity (Berestycki)
  • Schramm-Loewner evolution and discrete analogs

  • Random planar curves and Schramm-Loewner evolutions (Werner)
  • Conformally Invariant Processes in the Plane: Summer School Lecture Notes (Lawler)
  • Conformally Invariant Processes in the Plane: Book (Lawler --- save and use online ps2pdf if your machine doesn't have postscript).
  • A Guide to Stochastic Loewner Evolution and its Applications (Kager and Nienhuis)
  • Lectures on Schramm-Loewner evolution (Berestycki and Norris)
  • Growth models

  • Diffusion limited aggregation (Witten and Sander)
  • DLA bounds (Kesten)
  • Dielectric breakdown model (Niemeyer, Pietronero, Weismann)
  • Introduction to KPZ (Quastel)
  • Renormalization fixed point of the KPZ universality class (Corwin, Quastel, Remenik)
  • KPZ equation and universality class (Corwin)
  • Directed polymers (Borowin, Corwin, Ferrari)
  • Yang Mills

  • Quantum Yang-Mills Theory (Jaffe and Witten)
  • Large N lattice Gauge Theory (Chatterjee)
  • Large N master field in 2D (Levy)
  • Selected references on universal object relationships

    GFF + SLE

  • Imaginary Geometry I: Interacting SLEs (Miller and Sheffield; see also parts II, III, IV)
  • A contour line of the continuum Gaussian free field (Schramm and Sheffield)
  • LQG + LQG = LQG + SLE

  • Conformal weldings of random surfaces: SLE and the quantum gravity zipper (Sheffield)
  • CRT + CRT = LQG + SLE

  • Liouville quantum gravity as a mating of trees (Miller and Sheffield)
  • LQG + reshuffled SLE = LQG + DBM

  • Quantum Loewner Evolution (Miller and Sheffield)
  • LQG = TBM

  • Liouville quantum gravity and the Brownian map (Miller and Sheffield)

    Open problems:

    Open problem document in progress

    Problem sets and final project:

    Problem set document