Super-character theory and comparison arguments for a random walk on the upper triangular matrices

A Super-class Walk on Upper-triangular Matrices

Let G be the group of n×n upper-triangular matrices with elements in a finite field and ones on the diagonal. This paper applies the character theory of Andre, Carter and Yan to analyze a natural random walk based on adding or subtracting a random row from the row above.

Random walk on upper triangular matrices mixes rapidly

We present an upper bound O(n2) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant, and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process.

A SUPER - CLASS WALK ON UPPER - TRIANGULAR MATRICESEry

Two Random Walks on Upper Triangular Matrices

We study two random walks on a group of upper triangular matrices. In each case, we give upper bound on the mixing time by using a stopping time technique.

Mixing of the Upper Triangular Matrix Walk

We study a natural random walk over the upper triangular matrices, with entries in the field Z2, generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields Zq for q prime.