Upper triangular matrix walk: Cutoff for finitely many columns

Abstract

We consider random walk on the space of all upper triangular matrices with entries in double-struckF2 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.

The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$

“…The projection onto the final column of the matrix is itself a well known Markov chain known as the East Model. Ganguly, Lubetzky and Martinelli [11] proved that the East model exhibits cutoff and later Ganguly and Martinelli [12] extending this to the last k columns of the upper triangular matrix walk.…”

Spectral distributions of periodic random matrix ensembles

“…By now many works in probability have studied Markov chains on G(n, q) and their convergence to the stationary distribution, see for instance Stong [21], Coppersmith-Pak [10], Aris-Castro-Diaconis-Stanley [1], Peres-Sly [20], Ganguly-Martinelli [14], Nestoridi [19]. From the algebraic side, the character theory and conjugacy classes of these groups are 'wild type' and intractable, though various results exist; a nice exposition of these issues and of G(n, q) in general is given in Diaconis-Malliaris [11].…”