# Network questions in lilypond protocol models

*Stochastic Processes and Financial Mathematics*

*by Professor Daryl Daley*

*Institution:*Department of Mathematics and Statistics, The University of Melbourne

*Date: Thu 27th August 2009*

*Time: 3:15 PM*

*Location: Old Geology Theatre 2, The University of Melbourne*

*Abstract*: The germ-grain model for lilypond leaves has spawned a variety of models in other geometric settings. The simplest entails Poisson distributed points (`germs') on the line, and it leads to algebraically tractable answers to a range of questions. Most questions in other settings are amenable to solution only via simulation or sometimes via inequalities. The talk uses `classic' models in 1, 2 and 3-D, and grains that are finite line-segments in the plane, to illustrate the discussion.

Explicit constructions describe the growth of grains about the germs as centres until the growth is stopped by contact of the growing grain with another which in turn may have already stopped or may indeed keep growing. The resulting model can be described algebraically, which enables us to address the basic question of whether or not there exists an infinite `chain' of touching grains; call the existence of such `percolation'. When does a given model percolate? If it does not, what is the size of the finite clusters, and how much overlap (Penrose's `jumping frogs') results in percolation? What is the grain-size distribution? What is the distribution of the number of grains touching a given grain? What is the role of the Poisson distribution? What is the role of different granular shapes? Are there properties that persist across finite dimensions?

*For More Information:* contact: Prof Daniel Dufresne. email dufresne@unimelb.edu.au OR Dr Aihua Xia. email xia@ms.unimelb.edu.au