Percolation Theory 

(Elective Course in Probability for the M. Stat. Second Year and Ph.D. First Year)

Academic Year 2021-2022 : Semester I

Instructor : Rahul Roy

Teaching Assistant : Partha Pratim Ghosh

Course Outline : 

• Introduction to bond and site percolation. Formal probability set-up. Critical phenomenon and its existence.

• FKG inequality, BK inequality (only for increasing events), Russo’s formula..

• Exponential decay of the percolation probability below criticality.

• Uniqueness of the infinite open cluster.

• Critical probability for two dimensions is ½.

• Oriented percolation in two dimensions: Subadditive ergodic theory; introduction and the model; characterisation of p < pc . Recurrence properties of the right edge process. Exponential estimates for p < pc . Proof of no percolation at criticality. Exponential decay of time of ‘extinction’.

• Continuum percolation: The Boolean model; Coupling and scaling, FKG inequality; Occupancy in Boolean models; Vacancy in Boolean models, the covered volume fraction.
  
  

Reference Texts :

1. G. Grimmett, Percolation 2nd edition, Springer (1999).

2. T. Liggett, Interacting Particle Systems, Springer.

3. R. Durrett, Oriented percolation in two dimensions, Ann. Probab. 1984, 999-1040. 

4. R. Meester and R. Roy, Continuum Percolation, Cambridge (1996).