# 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