Upper triangular matrix walk: Cutoff for finitely many columns. Issue 2 (24th January 2019)

• Record Type: Journal Article
• Authors: Ganguly, Shirshendu; Martinelli, Fabio
• Is Part Of: Random structures & algorithms. Volume 55:Issue 2(2019)
• Page Start: 313
• Page End: 341
• Publication Date: 2019-01-24
• Subjects: East process -- Mixing time -- random walk on groups; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20838
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : We consider random walk on the space of all upper triangular matrices with entries in F 2 which forms an important example of a nilpotent group. Peres and Sly proved tight bounds on the mixing time of this walk up to constants. It is well known that the column projection of this chain is the one dimensional East process. In this article we complement the Peres‐Sly result by proving a cutoff result for the mixing of finitely many columns in the upper triangular matrix walk at the same location as the East process of the same dimension. The proof is based on a recursive argument which uses a local version of a dual process appearing in a previous study, various combinatorial consequences of mixing and concentration results for the movement of the front in the one dimensional East process.