The precise tail behavior of the total progeny of a killed branching random walk
Elie A\"id\'ekon, Yueyun Hu, Olivier Zindy

TL;DR
This paper precisely characterizes the tail distribution of the total progeny in a killed branching random walk, addressing an open problem and providing detailed asymptotic behavior in critical and subcritical regimes.
Contribution
It offers the first exact tail asymptotics for the total progeny of killed branching random walks in critical and subcritical cases, solving a longstanding open problem.
Findings
Derived precise tail distributions for total progeny
Solved an open problem posed by Aldous
Established asymptotic behaviors in critical and subcritical regimes
Abstract
Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · stochastic dynamics and bifurcation
