  We consider the localization game played on graphs in which a cop tries to
determine the exact location of an invisible robber by exploiting distance
probes. The corresponding graph parameter $\zeta(G)$ for a given graph $G$ is
called the localization number. In this paper, we improve the bounds for dense
random graphs determining an asymptotic behaviour of $\zeta(G)$. Moreover, we
extend the argument to sparse graphs.
