Geoffrey R. Grimmett Abstract. The square lattice is used to generate an oriented graph in which a rightward or upward arrow is present on each edge with probability a, and a leftward or downward arrow with probability b. Independence between different edges of the square lattice is assumed, but nothing is assumed concerning the dependence between the two possible orientations at any given edge...

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) ...