Four-regime variance conjecture for the chromatic number of random graphs

Let $a(n)$ and $x(n)$ be defined by $a(n)=\lfloor\alpha_0(n)-1/2\rfloor$ and

$$\mu_{a}(n)=\frac{2x n}{a^2}=\Theta\left(\frac{x n}{\log^2 n}\right).$$

For good $n$, let $g(n)$ be the fluctuation scale in the asymptotic normality conjecture, equivalently $g(n)=\sqrt{\operatorname{Var}(Y_n)}$, and set $c_0=2/\log 2$.

Four-regime variance conjecture. The following asymptotic estimates hold: (i) if $x\to0$, then

$$g(n)\sim\sqrt{\mu_a}\frac{\log\log n+\log(1/x)}{c_0\log^2 n};$$

(ii) if $x=\Theta(1)$, then

$$g(n)=\Theta\left(\sqrt{\mu_a}\frac{\log\log n}{\log^2 n}\right)=\Theta(w_n);$$

(iii) if $x\to\infty$ with $x=n^{o(1)}$, then

$$g(n)\sim\frac{\sqrt{\mu_a}}{x}\frac{\log\log n+\log x}{c_0\log^2 n}\sim c_1 n^{1/2}\frac{\log\log n+\log x}{\sqrt{x}\log^3 n};$$

and (iv) if $x\geqslant(\log n)^C$ for some constant $C>0$, then

$$g(n)=\Theta\left(\frac{\sqrt{\mu_a}\log x}{x\log^2 n}\right)=\Theta\left(\frac{\sqrt n\log x}{\sqrt{x}\log^3 n}\right).$$

These regimes refine the proposed width of the chromatic-number distribution. They cover all good $n$, with overlap between the third and fourth regimes; the transition constants are not determined in the middle regime, and the conjecture is open.