The sizes of the cycles and unicyclic components in the random graph $G(n, n/2\pm s)$, where $n^{2/3}\ll s \ll n$, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given: these distributions include the gamma distributions with parameters 1/4, 1/2 and 3/4, as well as the Poisson–Dirichlet and GEM distributions with parameters 1/4 and 1/2.