Tight concentration of star saturation number in random graphs

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{s}\mathrm{a}\mathrm{t}(G,F)$. For the star $F=K_{1,r}$, the asymptotics of $\mathrm{s}\mathrm{a}\mathrm{t}(G(n,p),F)$ is known. We prove a sharper result: whp $\mathrm{s}\mathrm{a}\mathrm{t}(G(n,p),K_{1,r})$ is concentrated in a set of 2 consecutive points.