Department Mathematik



                              Gibbsian Point Processes

                                            Winter semester 2022/23

This is a course for master students in Mathematics, Theoretical and Mathematical Physics, and Finance- and Insurance Mathematics. The course is egligible as Special topics in Stochastics as module WP 32 (for Math), as well as WP 43 and 44 (for TMP), resp. WP 10 (FIM, PO 2011) and WP 13 (FIM, PO 2019). Other moduls upon request.

Lecturer Dr. Matthew Dickson, Prof. Dr. Markus Heydenreich

Content: The initial focus is on a general theory of point process with special emphasize on the Poisson process. We then introduce continuum Gibbs measures, and discuss their properties. In the sequel we focus on more specialized topics (depending on time):

  • phase transitions
  • cluster expansions
  • spatial birth-death processes
The aim of the course is to make students acquainted with the foundations of continuum statistical mechanics. Ideal starting point for a master thesis in this area.

Lecture: Mon 10-12 and Wed 10-12 in lecture hall B047

Exercise: Weekly work sessions, Fri 14-16 in lecture hall B047

Registration: Sign up for this course via MOODLE, please (subscription key is "Gibbs").
All further communication is solely provided via moodle.

Exam: We plan an oral exam in February 2023.

We will loosely follow the lecture notes [Gibbsian Point Processes] by S. Jansen.
A key reference is the book [Lectures on the Poisson process] by G. Last and M. Penrose (CUP 2017).
Further references to the literature are provided during the course.

Note: During the winter semester, our group offers two courses on the master level, Stochastic Processes (SP) and Gibbsian Point Processes (GPP). Both courses are fairly independent and focus on different aspects. SP is a core module in probability theory, where many fundamental concepts are introduced, and it is therefore strongly advised to all students who plan to write a thesis in probability theory. GPP is of slightly different character, it presents a kaleidoscope of very specialized topics in a research area, and focusses in particular on the connection with (classical) statistical mechanics.