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Vorlesung + Übung: Advanced Discrete Probability (Fortgeschrittene Themen der diskreten Wahrscheinlichkeitstheorie) - Details
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General information

Course name Vorlesung + Übung: Advanced Discrete Probability (Fortgeschrittene Themen der diskreten Wahrscheinlichkeitstheorie)
Course number MTH-2700
Semester WS 2023/24
Current number of participants 18
Home institute Stochastik und ihre Anwendungen
Courses type Vorlesung + Übung in category Teaching
First date Thursday, 19.10.2023 15:45 - 17:15, Room: (1008/L)
Participants The course is open to Master students of Mathematics, Wi-Mathe, MRM, Data Science. Perfectly suited for Math + Computer Science track, also interested PhD students are welcome to attend.
Pre-requisites Probability (e.g. Stochastik I/II) + Linear Algebra.
Learning organisation Two lectures per week plus exercise session
Online/Digitale Veranstaltung Veranstaltung wird in Präsenz abgehalten.
Hauptunterrichtssprache englisch
Literaturhinweise [LPW] L. Levin, Y. Peres, E. Wilmers: Markov chains and mixing times. Second edition. AMS 2018.
[Gri] G. Grimmett: Probability on graphs. Second edition. Cambridge University Press 2018.
[LP] R. Lyons, Y. Peres: Probability on graphs and networks. Cambridge University Press 2016.
[DC] H. Duminil-Copin: Introduction to percolation theory. Lecture notes 2017.
[HH] M. Heydenreich, R. van der Hofstad: Progress in high-dimensional percolation and random graphs. Springer 2017.
[Hof] R. van der Hofstad: Random graphs and complex networks. Cambridge University Press 2017.
[Gri99] G. Grimmett: Percolation. Second Edition. Springer 1999.
[KP] J. Komjathy and Y. Peres: Topics in Markov chains: Mixing and escape rate.
Proceedings of Symposia in Pure Mathematics, Volume 91. AMS 2016.
[Häg] O. Häggström: Finite Markov Chains and Algorithmic Applicatons, LMS Student Texts 52, Cambridge University Press 2002.
Note: The sources [LPW], [Gri], [LP], [DC], [Hof] are available as linked pdf on author's homepages.

Rooms and times

No room preference
Tuesday: 15:45 - 17:15, weekly
(1008/L)
Thursday: 15:45 - 17:15, weekly (15x)
Friday: 10:00 - 11:30, weekly (15x)

Module assignments

Comment/Description

We discuss a number of special topics in the theory of discrete random structures. The lecture covers both classical foundations as well as very recent mathematical developments. The course is structures into five themes:

1) Markov chains on finite graphs, mixing time and cutoff phenomena
2) Random walk and electrical networks
3) Percolation Theory
4) Erdös-Renyi random graphs

A slight emphasis is on the third topic, Percolation Theory, which has been a prolific reserach field over the last two decennia. Among others, we will discuss Smirnov's celebrated proof of conformal invariance of the scaling limit of critical percolation in two dimension, and will get to know some of Duminil-Copin's elegant and short proofs.

There is no written syllabus, but I give references to all topics discussed during the lecture.