We estimate the expected mixing time of a random walk on a finite group supported by a random polylogarithmic set of elements. Following the spectral approach of Broder and Shamir, we present an alternative proof of the Dou-Hildebrand estimate and show that it holds almost surely. Good bounds on diameters follow from these results.
"On random random walks." Ann. Probab. 24 (2) 1001 - 1011, April 1996. https://doi.org/10.1214/aop/1039639375