Upper triangular matrix walk: Cutoff for finitely many columns

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.

Matrix Partitions with Finitely Many Obstructions

Each m by m symmetric matrix M over 0, 1, ∗, defines a partition problem, in which an input graph G is to be partitioned into m parts with adjacencies governed by M , in the sense that two distinct vertices in (possibly equal) parts i and j are adjacent if M(i, j) = 1, and nonadjacent if M(i, j) = 0. (The entry ∗ implies no restriction.) We ask which matrix partition problems admit a characteri...

Some rank equalities for finitely many tripotent matrices

‎A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations‎. ‎Moreover‎, ‎by using this equality and Theorems 8 and 10 in [Chen M‎. ‎and et al‎. ‎On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications‎, ‎The Scientific World Journal 2014 (2014)‎, ‎Article ID 702413‎, ‎7 page...

A SUPER - CLASS WALK ON UPPER - TRIANGULAR MATRICESEry

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.