Probability Study Group

This is an informal study group covering topics in probability. Speakers volunteer to present recent papers on topics selected by the group at large. Grad students, postdocs, and faculty are all encouraged to attend.

More details of Jeff Schenker's proof of eigenvector localization for band matrices

by Ben Rifkind | University of Toronto
Time: 13:00 — 14:30  (Monday, Apr. 26, 2010)
Location: Stewart Library, Fields Institute, 222 College St.
Abstract:
An N-by-N band matrix of band width W is a matrix that is identically zero outside a band of size W around the diagonal. Here we will consider the entries to be iid Gaussians. Jeff Schenker proved in a recent talk here that if $W^8/N ->0$, the eigenvectors are localized which means each eigenvector is concentrated in a vanishingly small set. This follows from showing that the resolvent R of this matrix satisfies the localization conditions $ER_{i,j} ~ e^{-C|i-j|/W^8}$ and $log(Var(R_{i,j}))$ is $O(|i-j|/W^8)$. This was done by coupling some independent randomness to a product structure of the resolvent and then extracting the large log variance from the independent randomness conditioned on the resolvent. This talk will explain these results in more detail.

Dates in this series

· Monday, Feb. 08, 2010: Liouville Property, Amenability, Discrete Fractals and Automaton Groups (Gidi Amir)
· Monday, Feb. 22, 2010: Scaling Exponents for Directed Polymers in 1+1 Dimensions, Part III (Tom Alberts)
· Monday, Mar. 01, 2010: Scaling exponents for directed polymers in dimension 1+1, part III (Tom Alberts)
· Monday, Apr. 26, 2010: More details of Jeff Schenker's proof of eigenvector localization for band matrices (Ben Rifkind)
· Monday, May. 10, 2010: The Doob h-transform: theory and examples (Alex Bloemendal)
· Monday, May. 17, 2010: Doob's h-transform: a few more examples (Alex Bloemendal)
· Monday, Oct. 04, 2010: Sample Path Properties of Stochastic Flows (Dima Dolgopyat)
· Monday, Oct. 18, 2010: Sample Path Properties of Stochastic Flows, Part II (Dima Dolgopyat)
· Monday, Oct. 25, 2010: Sample Path Properties of Stochastic Flows, Part III (Dima Dolgopyat)
· Friday, Jul. 29, 2011: SLE in multiply connected domains (Gregory F. Lawler)
· Friday, Dec. 02, 2011: A model of migration under constraints (Raoul Normand)
· Friday, Jan. 20, 2012: Local Limit Theorem for Random Walks on Free Groups (Andrew Stewart)
· Friday, Feb. 03, 2012: The Law of the Iterated Logarithm (Eric Hart)
· Friday, Mar. 16, 2012: Random matrices: Getting inside the bulk! (Van Vu)
· Friday, Mar. 30, 2012: A Brief Introduction to Compressed Sensing (Kai Yang)
· Friday, Apr. 13, 2012: The Minimum Cost of a Random Assignment (Mustazee Rahman)
· Tuesday, May. 01, 2012: An Introduction to the Dirichlet Process (Tom Alberts)