It has been conjectured since the work of Lalley and Sellke [26] that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida [8], Arguin et al. [3,4]). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. [4].

This paper deals with the large deviations behavior of a stochastic process called thinned Lévy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs [3]. The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Lévy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Lévy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

Consider a random walk in random environment on a supercritical Galton-Watson tree, and let τ n be the hitting time of generation n. The paper presents a large deviation principle for τ n /n, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time. Key words. Random walk in random environment, law of large numbers, large deviations, Galton-Watson tree.