Hi Gowers,

In this paper Sidoravicius, V. and Kesten, H . A shape theorem for the spread of an infection. Annals of Mathematics, v. 167, p. 1-63, 2008.

the authors consider a different model, but they were interested in the problem shape like you. One time Vladas told me that about a model similar what you describe and also that there is a believe about universality for those models, perhaps they discuss this on this reference.

This is a link for the arxiv version of this paper.

Hi Gowers,

In this paper: Sidoravicius, V. and Kesten, H . A shape theorem for the spread of an infection. Annals of Mathematics, v. 167, p. 1-63, 2008,

the authors consider a different model, but they were interested in the shape problem like you. One time before publish thi paper Vladas told me that about a model, similar to what you describe and also that there is a believe about universality for those models, perhaps they discuss this on this reference.

Here is a link for the arxiv version of this paper.