## Topic outline

• ### General

Note: The lecture hall for Wednesday's lectures has been changed to M3 (same as Monday's lectures).

##### Course description
Many complex systems in nature and society are composed of a large number of randomly interacting simple components. This course introduces you to mathematical methods for analyzing such systems, and shows how you can apply these methods to a wide range of stochastic models. The mathematical theory focuses on tightness and weak convergence of probability measures on large finite structures. Concrete examples and applications include random walks and Brownian motion, percolation and epidemics on graphs, Curie-Weiss model and Ising model, and voter model and contact process.

##### Contents

Theory:

• 0-1 laws
• Tightness and weak convergence of probability measures
• Couplings and monotonicity

Models and examples:

• Random walk and Brownian motion
• Percolation
• Curie-Weiss model and Ising model
• Voter model, contact process, and totally asymmetric exclusion process

##### Time
2018-2019 Period IV (6 weeks)
• Lectures: Mondays 10-12 and Wednesdays 10-12 in M3
• Exercises: Wednesdays 14-16 in Y228b

##### Prerequisites
Probability theory (MS-E1600) and Stochastic processes (MS-C2111), or equivalent.
The course grade is determined by a score consisting of
• up to 24 points from the exam (4 problems worth 6 points each)
• up to 6 bonus points from exercises

##### Exam
The exam will be held on:
• Wednesday 10.4.2019 at 13:00-16:00
There will be just one more time in 2019 to take the exam:
• Wednesday 29.5.2019 at 16:30-19:30
The exam consists of 4 problems, each worth 6 points. You are allowed to bring to the exam a handwritten memory aid sheet. The memory aid sheet must be of size A4 with text only on one side, and it must contain your name and student number in the upper right corner. You don’t need to return your memory aid sheet.

##### Exercises
There are weekly problem sets, posted under the "Assignments" tab on this page. Written solutions to the problems are to be returned to the course homework "mailbox" (on the first floor near Laskutupa Y190c) or as pdf files through MyCourses by Mondays at 10 am. For the grading of the course, these solutions amount to up to 6 additional points to the exam score.

In the exercise sessions on Wednesdays 14-16, the course teaching assistant will provide help in solving the problems. The exercise sessions may also contain brief recitations on the topics of the lectures and problems. You should think about the problems in advance, so as to be able to focus on whatever you find difficult during the time when the teaching assistant is there to help you!