Let k ⩾ 3 be a fixed integer and let Zk
(G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Zk
(G(n, m)) is known to be concentrated in the sense that
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,
converges to 0 in probability for any diverging function
. For k exceeding a certain constant k
0 this result covers all average degrees up to the so-called condensation phase transition
, 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’.