Schramm Loewner Evolution and Lattice Models

Fall 2021

Web page: http://www.math.toronto.edu/ilia/MAT1502.2021/.

Class Location & Time: Mon, 10:00 AM - 1:00 PM; Online.
No class on October 11 (Happy Canadian Thanksgiving!)
Extra class: November 29.

Instructor: Ilia Binder (ilia@math.toronto.edu).

Textbooks:

  1. Dmitry Beliaev, Conformal Maps and Geometry, World Scientific, 2019
  2. Gregory F. Lawler, Conformally Invariant Processes in the Plane, AMS, 2005
  3. Antti Kemppainen, Schramm-Loewner Evolution, Springer, 2017

Course notes:

  1. The introduction.
  2. Self Avoiding Random Walk: the definition.
  3. Parafermionic observable for Self Avoiding Random Walk.
  4. O(N) model.
  5. Potts model.
  6. Cardy-Smirnov observable for Critical percolation on hexagonal lattice.
  7. Conformal maps: distortion theory.
  8. Caratheodory convergence.
  9. The Extremal Length Method.
  10. Boundary behavior of conformal maps: first results.
  11. Prime ends.
  12. The Loewner Evolution.
  13. Loewner chains generated by slit domains.
  14. Chordal Loewner Evolution.
  15. The one dimensional Brownian Motion.
  16. Filtrations, adapted processes, stopping times.
  17. Conditional expectations and martingales.
  18. Maximal inequalities, optional stopping time theorem.
  19. Quadratic variation.
  20. Semi-martingales and Hardy spaces.
  21. Stochastic integration and Ito's formula.
  22. Levy characterization Theorem.
  23. Bessel Processes.
  24. SLE: an introduction.
  25. SLE: derivative estimates.
  26. SLE: phases.
  27. Dimension of SLE curves.
  28. SLE in other domains.
  29. Locality of SLE6.
  30. Radial SLE.
  31. Tightness in the families of Loewner curves.
  32. Critical Percolation: convergence of the interface.
  33. Polynomial rate of convergence to SLE.
  34. Gaussian Free Field.