Speed and fluctuations of N-particle branching Brownian motion with spatial selection

Abstract

We consider branching Brownian motion on the real line with the following selection mechanism: Every time the number of particles exceeds a (large) given number $N$, only the $N$ right-most particles are kept and the others killed. After rescaling time by ${\log }^{3}N$, we show that the properly recentred position of the $\lceil \alpha N\rceil$-th particle from the right, $\alpha \in (0,1)$, converges in law to an explicitly given spectrally positive Lévy process. This behaviour has been predicted to hold for a large class of models falling into the universality class of the FKPP equation with weak multiplicative noise [Brunet et al., Phys. Rev. E \textbf{73}(5), 056126 (2006)] and is proven here for the first time for such a model.

Continues and essentially replaces arXiv:1112.0266v1. Based on Chapter 2 of my PhD thesis at Université Pierre et Marie Curie, Paris, available at arXiv:1210.3500. Changes in v2 (74 pages): Reorganisation, simplifications in some places, typos corrected. Changes in v3 (84 pages): Many small corrections and additional details. Changes in v4 (87 pages): journal version, minor modifications

References in corpus (5)

• The genealogy of branching Brownian motion with absorption
• Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations
• The survival probability of a branching random walk in presence of an absorbing wall
• The number of absorbed individuals in branching Brownian motion with a barrier
• Survival of near-critical branching Brownian motion

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• A multi-state voter model with imperfect copying
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• Branching random walk with non-local competition
• A free boundary problem with non local interaction
• Cross-overs of Bramson's shift at the transition between pulled and pushed fronts
• Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models
• Negative large deviations of the front velocity of $N$-particle branching Brownian motion
• Emergent population dynamics of random walkers with cooperative reproduction and spatial selection
• Freezing in the Infinite-Bin Model