# Research in Scientific Computing in Undergraduate Education

### Multi-type branching processes with dependent offspring

The general theory of branching processes has been generalized to branching processes in which the population comprises multiple types (Mode, 1971). Multi-type branching processes have recently been considered in which offspring are potentially dependent (Gonzalez, Martinez & Mota, 2006).

Such processes arise already in relation to percolation on the integer lattice $\mathbb{Z}\times \mathbb{Z}$. To see how this occurs, consider a situation in which each bond in $\mathbb{Z}\times \mathbb{Z}$ is open with probability $p$. We regard the site $(0,0)$ as a base-point and look at geodesics (= shortest-length paths) of bonds from the base-point to another site. The length of a geodesic is the number of bonds comprising the geodesic:

Geodesic of length 4

If you prefer the language of graphs to that of percolation theory, replace “bond” with “edge” and “site” with “vertex”.

The geodesics come in three types:

#### Type 0

The unique geodesic of type 0 is the empty geodesic of length 0.

#### Type 2

There is an obvious notion of offspring of a geodesic – a geodesic $\gamma_1$ of length $n$  has as offspring those geodesics $\gamma_2$ of length $n+1$ that contain $\gamma_1$ as sub-paths. Following the model of percolation, we regard the bonds as open with probability $p$, and correspondingly, regard the geodesics as giving birth to offspring with probability $p$. The unique geodesic of length 0 gives birth to 4 geodesics of type 1, each geodesic of type1 gives birth to 1 geodesic of type 1 and 2 geodesics of type 2, and each geodesic of type 2 gives birth to 2 geodesics of type 2, all with probability $p$. The birth of offspring geodesics is not independent:

Dependent geodesics of length 3

The blue and red geodesics shown are not independent when $0: $\mathbb{P}(\textrm{red}\vert \textrm{blue})=p^2\neq p^3 =\mathbb{P}(\textrm{red})$.

The first aim of this project is to establish computational evidence for, and determine general conditions under which, there is a phase transition from extinction with probability 1 to extinction with probability 0 in multi-type branching processes with dependent offspring. We know that this will happen at $p = \frac{1}{2}$ for the integer lattice, because that is the percolation threshold. Can we ascertain this from general facts about multi-type branching processes with dependent offspring?