In this paper we study the random walk on the hypercube (ℤ / 2ℤ)n which at each step flips k randomly chosen coordinates. We prove that the mixing time for this walk is of the order (n / k)logn. We also prove that if k = o(n) then the walk exhibits cutoff at (n / 2k)logn with window n / 2k.

This note is motivated by Blom's work in 1989. We consider a generalized Ehrenfest urn model in which a randomly-chosen ball has a positive probability of moving from one urn to the other urn. We use recursion relations between the mean transition times to derive formulas in terms of finite sums, which are shown to be equivalent to the definite integrals obtained by Blom.

We consider a stochastic process in a modified Ehrenfest urn model. The modification prescribes there to be a minimum number of balls in each urn, and the process records the differences between treatment assignments under a sampling scheme implemented with this modified Ehrenfest urn model. In contrast to the result that the difference process forms a Markov chain and converges to a stationary distribution under the Ehrenfest urn model, the corresponding process under this modified Ehrenfest urn design satisfies the central limit property. We prove this asymptotic normality property using a central limit theorem for dependent random variables, renewal theory, and two Kolmogorov-type maximal inequalities.