Planting colourings silently.

Let k >= 3 be a fixed integer and let Z(k) (G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Z(k) (G(n, m)) is known to be concentrated in the sense that 1/n (lnZ(k) (G(n, m)) - lnE[Z(k) (G(n, m))]) converges to 0 in probability (Achlioptas and Coja-Oghlan, Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, 1/omega (lnZ k (G(n, m))lnE[ Z(k) (G(n, m))]) converges to 0 in probability for any diverging function omega - omega(n) -> infinity. For k exceeding a certain constant k(0) this result covers all average degrees up to the so-called condensation phase transition d k, cond, and this is best possible. As an application, we show that the experiment of choosing a k-colouring of the random graph G(n, m) uniformly at random is contiguous with respect to the so-called 'planted model'.