[Matematikcentrum] [Lunds universitet]

# NAMS005 (7.5 hp): PhD course on Percolation Theory and Random Graphs

Official syllabus: English, Swedish,

Current information (last updated: 12 April 2019):
• Next two lecture: 24 April; 2 May 2019
• The 25 April lecture is replaced by the lecture on the 24 April, in MH:228, time: 1015-1200
 Lectures: Every Thursday 1315-1500 (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, ℤ1 2 (14/3) Bond and site percolation on ℤ2, proof of monotonicity of θ(p), lower and upper bound for pc(Z2,bond), exact values, pc(ℤd,bond) for d≥3, monotonicity, asymptotics, equality of the critical points pc=pM, lower bound for pc(ℤ2,site) via method of generations. 3 (21/3) Estimation of pcsite(ℤ2) using multi-type branching process; matching graphs; Menshikov-Pelekh method. Continuity of θ(p) - concecture at pc (on ℤd: Kesten d=2; Slade-Hara d≥19; long-range d≥4). 4 (28/3) Increasing functions and events. Exploration process. Theorem: pc(bond)≤ pc(site)≤1-(1-pc(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 ℤ2 and 𝕋 7 (24/4) ... 8 (2/5) ...