Student Probability Seminar

Maximum of branching random walk

Speaker: Chris Thornett, CIMS

Location: Warren Weaver Hall 512

Date: Wednesday, October 31, 2018, noon


Imagine a particle who takes a step from some distribution, then
dies and gives birth to two children. These children then each take
another step, independent of one another, with the same distribution as
their parent, and then also die and give birth to two children of their
own. The process then continues in this fashion. After $n$ steps, there
are $2^n$ particles. A reasonable question is: what is the height of the
maximum one?

Clearly, the position of each particle has the same distribution, and any
one particle is unlikely to be far from its mean. However, there are so
many particles that it seems likely that the maximum has strictly larger
order than a "typical" particle. In this talk, I will derive the correct
order, and discuss the fluctuations (i.e. the maximum is still random, so
it would be helpful to know how accurate our estimate is). If time
permits, I will introduce some variations and discuss their similarities
and differences, such as BRW on Galton-Watson trees, and Branching
Brownian Motion.