Stable Point Processes, Branching Random Walks and a Prediction of Brunet and Derrida

Abstract: Stable point processes were introduced and characterized by Davydov, Molchanov and Zuyev (2008). They showed that such a point process can always be represented as a scale mixture of iid copies of one point process with the scaling points coming from an independent Poisson random measure. We obtain such a point process as the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. In particular, we show that a prediction of two statistical physicists, Brunet and Derrida (2011), remains valid in this setup, and recover a slightly improved version of a result of Durrett (1983) (this talk is based on a joint work with Ayan Bhattacharya and Rajat Subhra Hazra).