On the phase transition in certain percolation models.

  • Christopher Daniels

Student thesis: Doctoral ThesisPhD

Abstract

Consider random sequential adsorption on a chequerboard lattice with arrivals at rate 1 on light squares and at rate λ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites black, and the remaining squares white. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability p; otherwise allow diagonal connections between white squares.We show that there is a critical surface of pairs (λ, p), containing the pair (1,0.5), such that for (λ, p) lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates.We find conditions satisfied by a broad class of essentially planar percolation models such that for a model satisfying the conditions, the presence or absence of percolation is determined by what happens in a collection of finite boxes. This criterion applies to a (non-degenerate) Poisson Boolean model, to the random connection model for some sufficiently high p<1, and to the model described above.We also find conditions that do not require rotation invariance which produce a comparable result; these conditions seem plausible, but finding non-trivial examples is a matter for further research.
Date of Award22 Feb 2016
LanguageEnglish
Awarding Institution
  • University of Bath
SponsorsEngineering and Physical Sciences Research Council
SupervisorMathew Penrose (Supervisor) & Karsten Matthies (Supervisor)

Cite this

On the phase transition in certain percolation models.
Daniels, C. (Author). 22 Feb 2016

Student thesis: Doctoral ThesisPhD