Planting colourings silently

TL;DR

This paper proves a strong concentration result for the number of proper k-colorings in random graphs, extending previous work and showing that the distribution is tightly concentrated around its expectation for a wide range of degrees.

Contribution

It establishes a significantly stronger concentration of the number of k-colorings in random graphs, valid up to the condensation phase transition, and demonstrates contiguity with the planted model.

Findings

Strong concentration of Z_k(G(n,m)) around its expectation.

Valid for a wide range of average degrees up to the phase transition.

Shows contiguity between uniform coloring and the planted model.

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".