We show that the independence number of $ G_{ n,m}$ is concentrated on two values for $ n^{ 5/4+ \epsilon} < m \le \binom{ n}{ 2}$. This result establishes a distinction between $G_{ n,m}$ and $G_{ n,p}$ with $p = m/ \binom{ n}{ 2}$ in the regime $ n^{ 5/4 + \epsilon} < m< n^{ 4/3}$. In this regime the independence number of $ G_{ n,m}$ is concentrated on two values while the independence number of $ G_{ n,p}$ is not; indeed, for $p$ in this regime variations in $ \alpha( G_{ n,p})$ are determined by variations in the number of edges in $ G_{ n,p}$.
Categories: math.CO math.PR