关于随机树上λ-随机游动速度的一个注记

A note on the speed of a random walk on Galton-Watson trees

考虑随机树上λ-随机游动.现已知其速度存在并且不超过(m-λ)/(m+λ),其中m是平均子代数.若进一步考察速度与子代分布的关系,所有例子表明速度关于方差是单调的.常返概率是与速度密切相关的指标.本文讨论一族子代分布,其平均子代数m是固定的,证明了常返概率具有某种单调性,与预设的目标一致,更多相关问题有待进一步研究.

Abstract Consider a λ-biased random walk on Galton-Watson trees. It is proved that the speed exists and is bounded above by （m -λ）/（m ＋ λ）, where is is the mean of offsprings. One may further explore the relation between the speed and the offspring distribution. All examples show that the speed is a monotone function of the variance. We confirm this belief by verifying that the recurrent probability, a quantity related to the speed, is a monotone function of the variance of the offspring distribution in some sense, for the fixed m. Some observations are made and some questions are raised.