# Student Probability Seminar

#### Maximum of branching random walk

**Speaker:**
Chris Thornett, CIMS

**Location:**
Warren Weaver Hall 512

**Date:**
Wednesday, October 31, 2018, noon

**Synopsis:**

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.