## Biased voter model

Williams and Bjerknes (1972) introduced this
system as a model of skin cancer.
As in Richardson's model each site is either occupied (in state 1)
or vacant (in state 0). However this time

(i) An occupied site becomes occupied at a rate *delta* times the
fraction of the four nearest neighbors that are occupied.

(ii) A vacant site becomes occupied at a rate equal to the fraction of
the four nearest neighbors that are occupied.

If take *q = 0* then this reduces to Richardson's model. If
*q > 0* then we have the new complication that
if we start with only the origin occupied at time 0 then
there is positive probability the system will die out.

**Math Exercise. ** Show the probability of dying
out when you start with *k* occupied sites is *delta*^k
(i.e., *delta* to the *k*th power.

We can again start with B(0) = a finite set, say a small rectangle,
let B(t) be the set of lattice points occupied at time t ask:

**At what rate does the blob B(t) grow?**

If B(t) is ever the emptyset then it will remain so for all time.
In this case we say the model **dies out**.
Bramson and Griffeath (1980, 1981) showed (for a sketch of
their proof see Durrett (1988))

**Theorem.** If B(t) does not not die out,
then B(t)/t has a limiting shape.

**s3 Exercise.** Check the last theorem by setting *delta*=0.5
running the model starting from a 5 by 5 occupied rectangle.

Williams, T. and Bjerknes, R. (1972) Stochastic model for
abnormal clone spread through epithelial basal layer.
*Nature* **236**, 19-21

Bramson, M. and Griffeath, D. (1980). On the Willimas-Bjerknes tumor
growth model, II. *Proc. Camb. Phil Soc.* **88**, 339-357

Bramson, M. and Griffeath, D. (1981). On the Willimas-Bjerknes tumor
growth model, I. *Ann. Prob.* **9**, 173-185

Durrett, R. (1988) *Lecture Notes on Particle Systems and Percolation.*
Wadsworth Pub. Co., Pacific Grove, CA

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