# Fall 2012

Instructor: Antar Bandyopadhyay (Email: antar (at) isid (dot) ac (dot) in Office: 208 Faculty Building).

Class Time: Mon Thu 14:30 - 16:30 in Class Room 24.

Instructor's Office Hours: Mon Thu 16:30 - 17:30

Course Duration: July 23 - November 9, 2012

Midterm Examination: September 18, 2012, Time: 10:00 - 12:00 hours, Place: Conference Hall, Administrative Block

Final Examination: November 12, 2012, Time: 10:00 - 13:00 hours, Place: Conference Hall, Administrative Block

Course Outline:

• Basic concepts and definitions: percolation, percolation function, open cluster.
• I.I.D. Bernoulli Bond Percolation Model on Integer Lattice. Critical probability, non-triviality of critical probability.
• Configuration space, coupling.
• Basic tools: Increasing events, FKG Inequality; Disjoint occurrence of events, BK Inequality; Pivotality, Russo's Formula; Inequalities of reliability theory. Sub-additive limit theorem of analysis, generalizations.
• Sub-Critical Phase: Exponential decay of the tail of the radius of an open cluster, finer asymptotic of the tail of the radius of an open cluster. Definition of pT and proof of pc = pT. Connectivity function and asymptotic behavior of the connectivity function.
• Super-Critical Phase: Uniqueness of the infinite open cluster. Continuity of the percolation probability. Asymptotic of the tail of the radius of a finite open cluster and asymptotic behavior of truncated connectivity function.
• Bond percolation in two dimensional integer lattice. pc = 1/2.
• Amenable and non-amenable graphs.
• Transitive graphs and Cayley graphs.
• Uniqueness of infinite cluster in super-criticality for connected amenable infinite transitive graphs.
• Definition and relations between pc and pu.
• Slab percolation criticality and pc = pcslab.
• Number of open clusters per vertex.
• Introduction to Fractal Percolation.

References:

• Percolation by G. Grimmett.
• Percolation by B. Bollobas and O. Riordan.

Prerequisites:

• Real Analysis (at the level of Principles of Mathematical Analysis by W. Rudin).
• Measure Theoretic Probability (at the level of Probability and Measure by P. Billingsley).
• Liner Algebra (at the level of Finite Dimensional Vector Spaces by P. R. Halmos).

• Assignments: 15% of the total credit.
• Surprise Tests: 5% of the total credit.
• Class Presentation: 10% of the total credit.
• Midterm Exam: 20% of the total credit.
• Final Exam: 50% of the total credit.

Suggested Topics and References for Class Presentations

• Site percolation: Section 1.6 and Section 3.4 of Grimmett.
• Number of open clusters per vertex: Chapter 4 of Grimmett.
• Slab percolation criticality and its limit in dimension three and above: Sections 7.1 and 7.2 of Grimmett.
• Exponential and Sub-Exponential decay of the cluster size distribution: Section 6.3 and Section 8.6 of Grimmett.
• Menshikov's proof of exponential tail decay of the radius of an open cluster in sub-critical regime: Section 5.2 of Grimmett.
• Fractal Percolation: Section 13.4 of Grimmett.
• Continuum percolation: Section 12.10 of Grimmett.

Assignment Policies:

• There will be a total of 5 sets of homework assignments each carrying a total of 30 points. 4 best assignment scores will be taken for the final grading.
• The assignments will be given in class on every alternate Monday, and it will be due in class on Monday after two weeks. Each assignment set will be based on course materials covered in the lectures in the week before and the week it is assigned. For example, the assignment given on August 06, 2012 (Monday) will be due on August 20, 2012 (Monday).
• Late submission of an assignment will not be accepted. If you can not submit an assignment on time, don't worry about it, and try to do well in the others. It will not count in your final grade since you have one extra assignment anyway.
• Graded assignments will be returned in the class on the Thursday following the due date. For example, the assignment which is due on August 20, 2012 (Monday) will be returned after grading on August 23, 2012 (Thursday).