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Branching random walk with selection at critical rate. (English) Zbl 1392.60070

Summary: We consider a branching-selection particle system on the real line. In this model, the total size of the population at time \(n\) is limited by \(\exp(an^{1/3})\). At each step \(n\), every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the \(\exp(a(n+1)^{1/3})\) rightmost children survive to form the \((n+1)\)th generation. This process can be seen as a generalisation of the branching random walk with selection of the \(N\) rightmost individuals, introduced by E. Brunet and B. Derrida [“Shift in the velocity of a front due to a cutoff”, Phys. Rev. E 56, No. 3, 2597–2604 (1997; doi:10.1103/physreve.56.2597)]. We obtain the asymptotic behaviour of position of the extremal particles alive at time \(n\) by coupling this process with a branching random walk with a killing boundary.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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