Percolation Theory (MStat 2nd Year)

Notes for Prof. Rahul Roy's course on Percolation theory offered at ISI, Delhi.

The videos from last year’s course on percolation theory can be found here.

Zoom ID: 94809490278. Password: 482380.

Goal of the course

  • Bond percolation on \(\mathbb{Z}^d\)
  • Orientable percolation
  • Continuum percolation

References

  • Percolation - Grimmet
  • Percolation - Bollobas and Riordan

NOTES:

  • Lecture 1, 25th July: Introduction, Comparison with the random walk, definition of \(p_c(d)\), monotonicity of \(\theta(p)\).
  • Lecture 2, 30th July: Non-triviality of phase transition for \(\mathbb{Z}^d, d \geq 2\). Peierls argument, proof of Fekete’s lemma. Harris-FKG inequality.
  • Lecture 3, 1st August: Intuition and proof of FKG inequality. Idea of BK inequality.
  • Lecture 4, 6st August: Inductive proof of the BK inequality. Russo’s formula and pivotality.
  • Lecture 5, 8th August: Proof of Russo’s formula. Applications to establish upper and lower bounds on \(\beta_{n+m} (p) := \mathbb{P}_p(C \cap \delta B_{n+m})\) in terms of \(\beta_{n}(p)\beta_{m}(p).\)
  • Lecture 6, 13th August: Defining the connective rate constant \(\phi(p)\). Asymptotic tail behaviour of the radius of an open cluster. Properties of \(\phi(p)\) and statement of Burton-Keane.
  • Lecture 7, 20th August: Basic ergodic theory. Proof of \(N_{\infty} \in \{0,1,\infty\}\) where \(N_{\infty} := {\text{number of infinite clusters}}\).
  • Lecture 8, 24th August: Encounter points and proof of Burton-Keane for \(\mathbb{Z}^d\).
  • Lecture 9, 27th August:Zhang’s argument and proof of Kesten’s theorem about the critical point of \(\mathbb{Z}^2.\)
  • Lecture 10, 3rd September: Decision trees, differential inequalities the OSSS inequality proof of subcritical sharpness.
  • Lecture 11, 5th September: Proof of OSSS inequality.
  • Lecture 12, 19th September: Defining the correlation length \(\xi(p)\). Connection to \(\phi(p)\). Proof that \(\xi(p) > \chi(p).\)
  • Lecture 13, 24th September: Percolation on subsets of \(\mathbb{Z}^2\).
  • Lecture 14: Ergodic Theory
  • Lecture 15,16,17,18: Orientable Percolation
  • Lecture 19,20,21: Continuum Percolation