arXiv subjects

Clyde H. Schoolfield, Jr.

Publications and source records attributed to Clyde H. Schoolfield, Jr..

3 recordsLinked to original sources

arXiv2000

Random walks on wreath products of groups

We bound the rate of convergence to uniformity for certain random walks on the complete monomial groups G \wr S_n for any group G. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group Z_2 \wr S_n, the generalized symmetric group Z_m \wr S_n, and S_m \wr S_n. These results provide benchmarks to which many other random walks, modeling a wide range of phenomena, may be compared using the comparison technique, thereby yielding bounds on the rates of convergence to uniformity for previously intractable random walks.

math.PR↗

arXiv2000

A signed generalization of the Bernoulli-Laplace diffusion model

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli-Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space (Z_2 \wr S_n) / (S_r \times S_{n-r}). Specifically, for r not too far from n/2, we determine that, to first order in n, 1/4 n \log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the ``cutoff phenomenon.''

math.PR↗

arXiv2000

Mixing times for Markov chains on wreath products and related homogeneous spaces

We develop a method for analyzing the mixing times for a quite general class of Markov chains on the complete monomial group G \wr S_n (the wreath product of a group G with the permutation group S_n) and a quite general class of Markov chains on the homogeneous space (G \wr S_n) / (S_r \times S_{n - r}). We derive an exact formula for the L^2 distance in terms of the L^2 distances to uniformity for closely related random walks on the symmetric groups S_j for 1 \leq j \leq n or for closely related Markov chains on the homogeneous spaces S_{i + j} / (S_i \times S_j) for various values of i and j, respectively. Our results are consistent with those previously known, but our method is considerably simpler and more general.

math.PR↗