# The lilypond model: existence, uniqueness and absence of percolation

*by Prof Gunter Last*

*Institution:*Uni Karlsruhe (Germany)

*Date: Fri 29th July 2005*

*Time: 3:15 PM*

*Location: Theatre 2, Ground Floor, ICT Building (111 Barry St, Carlton)*

*Abstract*: The lilypond system model based on a locally finite subset

$\varphi$ of the Euclidean space R^n is defined as follows. At time 0

every point of $\varphi$ starts growing with unit speed in all directions

to form a system of balls in which any particular ball ceases its growth

at the instant that it collides with another ball. A stochastic version of

this model has been introduced by HÃ¤ggstrÃ¶m and Meester (1996) in the

context of percolation and by Daley, Stoyan and Stoyan (1999) in the

context of hard-sphere models with maximal non-overlapping spheres.

In the first part of the talk we will present recent results obtained

jointly with Matthias Heveling (Karlsruhe) showing that the lilypond model

is uniquely defined for any locally finite subset $\varphi$. Actually we

will show that these results apply in the far more general setting, where

$\varphi$ is a locally finite subset of some pseudo-metric space. We will

also discuss an algorithm approximating the system with at least linearly

decreasing error.

In the second part of the talk we will consider a stochastic lilypond

model based on a stationary point process N. We present analytic

conditions on N implying the absence of percolation in this model.

Examples are Cox, Poisson cluster and Gibbs processes satisfying certain

exponential moment conditions. This part of the talk is based on joint

work with Daryl Daley (Canberra).

*For More Information:* Emma Lockwood tel: 8344-1617 email: emmal@ms.unimelb.edu.au