We present an analytical calculation of the extreme value statistics for dark matter haloes – i.e., the probability distribution of the most massive halo within some region of the universe of specified shape and size. Our calculation makes use of the counts‐in‐cells formalism for the correlation functions, and the halo bias derived from the Sheth–Tormen mass function. We demonstrate the power of the method on spherical regions, comparing the results to measurements in a large cosmological dark matter simulation and achieving good agreement. Particularly good fits are obtained for the most likely value of the maximum mass and for the high‐mass tail of the distribution, relevant in constraining cosmologies by observations of most massive clusters.

We consider the Gumbel or extreme value statistics describing the distribution function pG(νmax) of the maximum values of a random field ν within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of pG which is expected to hold in a regime where local maxima of the field are moderately high and weakly clustered. When the patch size becomes sufficiently large, the negative of the logarithm of the cumulative extreme value distribution is simply equal to the average of the Euler characteristic of the field in the excursion ν≥νmax inside the patches. The Gumbel statistics therefore represents an interesting alternative probe of the genus as a test of non‐Gaussianity, e.g. in cosmic microwave background temperature maps or in 3D galaxy catalogues. It can be approximated, except in the remote positive tail, by a negative Weibull‐type form, converging slowly to the expected Gumbel‐type form for infinitely large patch size. Convergence is facilitated when large‐scale correlations are weaker. We compare the analytic predictions to numerical experiments for the case of a scale‐free Gaussian field in two dimensions, achieving impressive agreement between approximate theory and measurements. We also discuss the generalization of our formalism to non‐Gaussian fields.

We explore the behaviour of accreting protoclusters with a Monte Carlo dynamical code in order to evaluate the relative roles of accretion, two‐body relaxation and stellar collisions in the cluster evolution. We corroborate the suggestion of Clarke and Bonnell that the number of stellar collisions should scale as (independent of other cluster parameters, where N is the number of stars in the cluster and the rate of mass accretion) and thus strengthen the argument that stellar collisions are more likely in populous (large N) clusters. We however find that the estimates of Clarke and Bonnell were pessimistic in the sense that we find that more than 99 per cent of the stellar collisions occur within the post‐adiabatic regime as the cluster evolves towards core collapse, driven by a combination of accretion and two‐body relaxation. We discuss how the inclusion of binaries may reduce the number of collisions through the reversal of core collapse but also note that it opens up another collisional channel involving the merger of stars within hard binaries; future N‐body simulations are however required in order to explore this issue.