The Zigzag Conjecture for the chromatic number of random graphs

Let $G_{n,p}$ denote the binomial random graph, let $\chi(G)$ be its chromatic number, and let $\theta=\theta(n)$ be defined by $\mu_\alpha=n^\theta$, where $\mu_\alpha$ is the expected number of independent sets of the relevant size $\alpha$. Set

$$\lambda=\max\left(\frac{\theta}{2},\frac{1-\theta}{2}\right).$$

Zigzag Conjecture. Set $p=\frac12$. There is a sequence of intervals of length $n^{\lambda+o(1)}$ containing $\chi(G_{n,1/2})$ with high probability. However, for every fixed $\varepsilon>0$ and every sequence of intervals $I_n$ of length $n^{\lambda-\varepsilon}$,

$$\mathbb{P}\left(\chi(G_{n,1/2})\in I_n\right)=o(1).$$

This conjecture identifies the concentration width with the larger of the fluctuations arising from independent sets of sizes $\alpha$ and $\alpha-1$. The paper suggests analogous statements for other constant edge probabilities, but the stated conjecture concerns $p=1/2$ and the case where $\theta(n)$ stays bounded away from $0$ and $1$.