[Matematikcentrum]
[Lunds universitet]
Official syllabus: English, Swedish,
Lectures: | Every Thursday 13^{15}-15^{00} (except 18/4 and 25/4) |
Room: | MH:227 |
Texts: Grimmett's book online; another Grimmett's book, van der Hofstad notes, more to come...
Remaining topics: Method of projections, Menshikov's theorem, Continuous, bootstrap, and AB percolation ...
Schedule:
Lectures | Subject |
1 (7/3) | Introduction, review of relevant topics from probability theory: 0-1 law, branching processes, Borel-Cantelli. Basic definitions, site/bond percolation, relation; infinite cluster, usual questions. Example: binary tree, ℤ^{1} |
2 (14/3) | Bond and site percolation on ℤ^{2}, proof of monotonicity of θ(p), lower and upper bound for p_{c}(Z^{2},bond), exact values, p_{c}(ℤ^{d},bond) for d≥3, monotonicity, asymptotics, equality of the critical points p_{c}=p_{M}, lower bound for p_{c}(ℤ^{2},site) via method of generations. |
3 (21/3) | Estimation of p_{c}^{site}(ℤ^{2}) using multi-type branching process; matching graphs; Menshikov-Pelekh method. Continuity of θ(p) - concecture at p_{c} (on ℤ^{d}: Kesten d=2; Slade-Hara d≥19; long-range d≥4). |
4 (28/3) | Increasing functions and events. Exploration process. Theorem: p_{c}^{(bond)}≤ p_{c}^{(site)}≤1-(1-p_{c}^{(bond)})^{ⅆ}. Proof of the continuity of θ(p) . |
5 (4/4) | Uniquenes of the infinite cluster, Harris-Fortuin-Kasteleyn-Ginibre inequality |
6 (11/4) | BK inequality, Morgulis-Russo's formula. Strict inequalities for the critical percolation threshold on ℤ^{2} and 𝕋 |
7 (24/4) | ... |
8 (2/5) | ... |
Examination: Please contact the lecturer, Stanislav Volkov