### MS-E1602 - Large Random Systems D, 01.03.2021-07.04.2021

Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.

#### LEARNING OUTCOMES

After completing the course the participant is able to:

- Formulate mathematical models of various systems with a large number of interacting random components

- Incorporate spacial structure and dynamics into probabilistic models

- Estimate the asymptotics of probabilities and expected values in models with a size parameter

- Formulate qualitative phase transitions in stochastic models and recognize them

- Verify if a sequence of probability distributions on a metric space converges

Credits: 5

Schedule: 01.03.2021 - 07.04.2021

Teacher in charge (valid 01.08.2020-31.07.2022): Kalle Kytölä

Teacher in charge (applies in this implementation): Kalle Kytölä

Contact information for the course (valid 08.02.2021-21.12.2112):

Lecturer: Kalle Kytölä

Teaching assistant: Osama Abuzaid

CEFR level (applies in this implementation):

Language of instruction and studies (valid 01.08.2020-31.07.2022):

Teaching language: English

Languages of study attainment: English

##### Content
• Valid 01.08.2020-31.07.2022:

Stochastic models with spatial and temporal structure

- 0-1 laws

- Large deviation estimates of rare events

- Phase transitions in stochastic models

- Convergence and tightness of probability measures

• Applies in this implementation:

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

##### Assessment Methods and Criteria
• Valid 01.08.2020-31.07.2022:

Weekly exercises and exam.

• Applies in this implementation:

The assessment is based on exercises and exam, but the details in remote mode are yet to be fixed.

• Valid 01.08.2020-31.07.2022:

- 2 x 2h lectures

- 1 x 2h exercises sessions

#### DETAILS

##### Study Material
• Valid 01.08.2020-31.07.2022:

Kemppainen and Kytölä: Large Random Systems, lecture notes, 2019.

• Applies in this implementation:

##### Prerequisites
• Valid 01.08.2020-31.07.2022:

MS-E1600 Probability Theory