Let G be a quasirandom graph on n vertices, and let W be a random walk on G
of length alpha n^2. Must the set of edges traversed by W form a quasirandom
graph? This question was asked by B\"ottcher, Hladk\'y, Piguet and Taraz. Our
aim in this paper is to give a positive answer to this question. We also prove
a similar result for random embeddings of trees.