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Upper triangular matrix walk: Cutoff for finitely many columns | Shirshendu Ganguly
; Fabio Martinelli
; | Date: |
27 Dec 2016 | Abstract: | We consider random walk on the group of uni-upper triangular matrices with
entries in $mathbb{F}_2$ which forms an important example of a nilpotent
group. Peres and Sly (2013) proved tight bounds on the mixing time of this walk
up to constants. It is well known that the single 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. Moreover, we also show that the spectral gaps of
the matrix walk and the East process are equal. The proof of the cutoff result
is based on a recursive argument which uses a local version of a dual process
appearing in Peres and Sly (2013), various combinatorial consequences of mixing
and concentration results for the movement of the front in the one dimensional
East process. | Source: | arXiv, 1612.8741 | Services: | Forum | Review | PDF | Favorites |
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