The Klee-Minty random edge chain moves with linear speed

An infinite sequence of 0's and 1's evolves by flipping each 1 to a 0 exponentially at rate 1. When a 1 flips, all bits to its right also flip. Starting from any configuration with finitely many 1's to the left of the origin, we show that the leftmost 1 moves right with bounded speed. Upper and lower bounds are given on the speed. A consequence is that a lower bound for the run time of the random-edge simplex algorithm on a Klee-Minty cube is improved so as to be quadratic, in agreement with the upper bound.