Randomly colouring simple hypergraphs

We study the problem of constructing a (near) random proper q-colouring of a simple k-uniform hypergraph with n vertices and maximum degree ∆. (Proper in that no edge is mono-coloured and simple in that two edges have maximum intersection of size one). We give conditions on q,∆ so that if these conditions are satisfied, Glauber dynamics will converge in O(n log n) time from a random (improper) start. The interesting thing here is that for k ≥ 3 we can take q = o(∆).