Yuval Peres, Microsoft Research

Title: Balanced self-interacting random walks

Date: June 4, 2:00-3:30

Abstract: It is well known that a random walk in d>2 dimensions where the steps are i.i.d. mean zero and fully supported (not restricted to a hyperplane), is transient. Benjamini, Kozma and Schapira asked if we still must have transience when each step is chosen from either μ1 or μ2 based on the past, where μ1 and μ2 are fully supported mean zero distributions. (e.g. we could use μ1 if the current state has been visited before, and μ2 otherwise). We answer their question, and show the answer can change when we have three measures instead of two. To prove this, we will adapt the classical techniques of Lyapunov functions and excessive measures to this setting. No prior familiarity with these methods will be assumed, and they will be introduced in the talk. Many open problems remain in this area, even in two dimensions. Lecture based on joint work with Serguei Popov (Campinas) and Perla Sousi (Cambridge).

Gordon Slade, University of British Columbia

Title: Integral representations for the self-avoiding walk

Date: June 8, 2:00-3:30

Abstract: The self-avoiding walk is a fundamental model in probability, combinatorics and statistical mechanics, for which many of the basic mathematical problems remain unsolved. Recent and ongoing progress for the four-dimensional self-avoiding walk has been based on a renormalization group analysis. This analysis takes as its starting point an exact representation of the self-avoiding walk problem as an equivalent problem for a perturbation of a Gaussian integral involving anti-commuting variables (fermions). This lecture will give a self-contained introduction to fermionic Gaussian integrals and will explain how they can be used to represent self-avoiding walks.

The talk is mainly based on the paper: D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys, 6:34--61, (2009)

Bálint Virág, University of Toronto

Title: The sound of random graphs

Date: June 15, 2:00-3:30

Abstract: Infinite random graphs, such as Galton-Watson trees and percolation clusters may have real numbers that are eigenvalues with probability one, providing a consistent "sound". These numbers correspond to atoms in their density-of-states measure.

When does the sound exist? When are there only finitely many atoms? When is the measure purely atomic? I will review many examples and show some elementary techniques for studying these problems, including some developed in joint works with Charles Bordenave and Arnab Sen. The last question is open for percolation clusters in Zd, d≥3, and for incipient Galton-Watson trees.

Alexander Holroyd, Microsoft Research

Title: Invariant matching

Date: June 22, 2:00-3:30

Abstract: Suppose that red and blue points occur as independent point processes in Rd, and consider translation-invariant schemes for perfectly matching the red points to the blue points. (Translation-invariance can be interpreted as meaning that the matching is constructed in a way that does not favour one spatial location over another). What is best possible cost of such a matching, measured in terms of the edge lengths? What happens if we insist that the matching is non-randomized, or if we forbid edge crossings, or if the points act as selfish agents? I will review recent progress and open problems on this topic, as well as on the related topic of fair allocation. In particular I will address some surprising new discoveries on multi-colour matching and multi-edge matching.

Rick Kenyon, Brown University

Title: Laplacians and connections

Date: June 25, 2:00-3:00

Abstract: We discuss the Laplacian operator on vector bundles on graphs, in particular relating its determinant to the enumeration of "cycle-rooted spanning forests" which are combinatorial objects generalizing spanning trees.