We study the extremes of a class of Gaussian fields with in-built hierarchical structure.The number of scales in the underlying trees depends on a parameter α ∈ [0, 1]: choosing α = 0 yields the random energy model by Derrida (REM), whereas α = 1 corresponds to the branching random walk (BRW). When the parameter α increases, the level of the maximum of the field decreases smoothly from the REM-to the BRWvalue. However, as long as α < 1 strictly, the limiting extremal process is always Poissonian.

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We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold h c for the magnitude of the external field necessary for trivialization and conjecture h c to be sharp. The Kac-Rice formula is our main tool. Our work extends Fyodorov [14], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.

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We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.