Planting colourings silently

Abstract

Let $k\geq3$ 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 $\frac1n(\ln Z_k(G(n,m))-\ln E[Z_k(G(n,m))])$ converges to $0$ in probability [Achlioptas and Coja-Oghlan: FOCS 2008]. In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, $\frac1\omega(\ln Z_k(G(n,m))-\ln E[Z_k(G(n,m))])$ converges to $0$ in probability for any diverging function $\omega=\omega(n)\to\infty$. 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".