STAT 260: Problems in Probability (Spring 2006)

Instructor Yuval Peres

Class time Tuesday and Thursday, 2:00 - 3:30 (room 332 Evans Hall).

Lecture notes

Course description

More detailed Course Topics

• The minimal spanning tree with random weights: open problems in the lattice and in the complete graph.
• The Ising model and the random cluster model
• Broadcasting on trees, and its interpretations in mathematical genetics and the Ising model. Determining critical values for reconstruction in larger alphabets is open.
• Markov chain mixing and the cutoff phenomenon.
• Glauber dynamics for the Ising model: dependence on temperature.
• Recursions on trees and nonlinear capacity.
• First-passage percolation and branching random walk.
• Random walks on groups, amenability and boundaries.
• An introduction to point processes; open problems on matchings and allocations for point processes.
• Domination and equipolarity between trees; Sidorenko's conjecture on bipartite graphs.

Prerequisite

Grading Students taking the course for credit are expected to

• Write up one lecture, with suitable references, in consultation with the instructor.
• Read a short research paper and present it in class (25 minutes). Suggestions will be provided.

Office Hours

Yuval Peres Tu and Th, 3:30-4:00 in 341 Evans.

Some suggested papers for student talks

Can be found e.g. on the homepages of

Itai Benjamini, Chris Hoffman, Russell Lyons, Laszlo Lovasz , Robin Pemantle, Yuval Peres ,

Oded Schramm, Alistair Sinclair, David Wilson, Balint Virag .