Range of cube-indexed random walk

For given finite, connected, bipartite graphG=(V,E) with distinguishedν ₀ ∈V, set {fx189-1} Our main result says there is a fixedb so that whenG is a Hamming cube ({0, 1} ⁿ with the usual adjacency), andf is chosen uniformly fromF, the probability thatf takes more thanb values is at most e^?Ω(n). this settles in a very strong way a conjecture of I. Benjamini, O. Häggström and E. Mossel [2]. The proof is based on entropy considerations and a new log-concavity result.