5 research outputs found
Transient Random Walks in Random Environment on a Galton-Watson Tree
We consider a transient random walk in random environment on a
Galton--Watson tree. Under fairly general assumptions, we give a sharp and
explicit criterion for the asymptotic speed to be positive. As a consequence,
situations with zero speed are revealed to occur. In such cases, we prove that
is of order of magnitude , with . We also
show that the linearly edge reinforced random walk on a regular tree always has
a positive asymptotic speed, which improves a recent result of Collevecchio
\cite{Col06}
Martingale ratio convergence in the branching random walk
We consider the boundary case in a one-dimensional supercritical branching random walk, and study two of the most important martingales: the additive martingale (Wn) and the derivative martingale (Dn). It is known that upon the system's survival, Dn has a positive almost sure limit (Biggins and Kyprianou [9]), whereas Wn converges almost surely to 0 (Lyons [22]). Our main result says that after a suitable normalization, the ratio Wn/Dn converges in probability, upon the system's survival, to a positive constant
The precise tail behavior of the total progeny of a killed branching random walk
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 its precise tail distribution both in the critical and subcritical cases, which solves an open problem of D. Aldous [4].
Keywords: Killed branching random walk; total progeny; spinal decomposition; Yaglom-type theorem; time reversed random wal
The precise tail behavior of the total progeny of a killed branching random walk
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 its precise tail distribution both in the critical and subcritical cases, which solves an open problem of D. Aldous [4]