Topic outline

  • Lecturer: Kalle Kytölä

    Teaching assistant: Osama Abuzaid

    Discussion forum: Zulip chat

    • This Zulip chat will be used for all course related discussions, both related to mathematics and to course arrangements (by contrast, this MyCourses page contains essentially static information). You can log in on this Zulip with your usual Aalto account. The data is hosted at Aalto University servers, but the policy is to not post any sensitive information in this discussion forum; see the general rules of
    • In the remote teaching mode, Zulip is the best platform I know of for actually interacting and discussing, both with the other participants and the teachers. I strongly encourage you to make the best use of it! --Kalle Kytölä (lecturer)

    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.



    • 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

    2020-2021 Period IV (6 weeks)
    • Lectures: Mondays 10-12 and Wednesdays 10-12 via Zoom
    • Exercises: Wednesdays 14-16 (?)

    Probability theory (MS-E1600) and Stochastic processes (MS-C2111), or equivalent.


    The course grade will be based on exercises and an adjustment of at most 15% based on an oral exam.

    In the end of the course we will hold 45 minutes oral exams for everyone at an individually scheduled time, where you are expected to explain one of your homework solutions (randomly chosen) and explain the topic and main result of one of the 12 lectures (randomly chosen) briefly. The grading is mainly based on exercises, but a small adjustment of at most 15% may be made based on the oral exam.

    There are weekly problem sets, posted under the "Assignments" tab on this page. Written solutions to the problems are to be returned as pdf files via the folder on the "Assignments" tab.