# 19208111 Masterseminar Stochastics

- FU-Students only need to register via Campus Management.
- Non-FU-students are required to register via MyCampus/Whiteboard.

### Winter Term 2022/2023

Lecturer: **Prof. Dr. Nicolas Perkowski**

**Time and place:**Tuesdays, 14--16h, SR 115, Arnimallee 3

If you want to participate, please write an email to perkowski@math.fu-berlin.de

**Prerequisites:** Stochastics I und II. No knowledge from physics or more advanced probability is needed!**Target group:** BMS students, Master students and advanced Bachelor students.

**Contents:** Contrary to its name, the seminar addresses both Bachelor and Master students and the subjects of most talks can serve as starting points for Bachelor theses. There will also be talks on more advanced subjects that can lead to Master theses.

We will study lattice models from statistical mechanics, with a particular emphasis on phase transitions. The basic question in statistical mechanics is "How can we explain physics on the observable, macroscopic scale from the processes occurring on the molecular, microscopic scale?", and we will get a small glimpse into this field.

After discussing basic concepts of thermodynamics, we will focus mostly on simple microscopic stochastic lattice systems that serve as toy models for physical processes. By "zooming out" from the atomistic to the observable scale we can derive laws of thermodynamics as mathematical theorems, rather than postulating them.

Our guiding example will be the Ising model, a toy model for a magnet, and we will study its phase transition at the critical temperature: If a ferromagnet is heated above a critical temperature, it loses its magnetic properties. But we will also discuss a simpler mean field approximation (Curie-Weiss model). Further subjects could be lattice gases (a toy models for ideal gases), percolation (a toy models for a porous medium), or, more advanced, interacting particle systems.

### Talks

Date |
Subject |
Speaker |

18.10. | - | - |

25.10. | Thermodynamics and Statistical Mechanics, Discussion of subjects | Nicolas Perkowski |

01.11. | - | - |

08.11. | The Curie-Weiss model | Luc Schoenmakers |

15.11. | - | - |

22.11. | Stein's method for concentration inequalities, applications to Curie-Weiss and maybe Ising | Ibraim Ibraimi |

29.11. | The Ising model: Existence of the pressure + magnetization, first definition of a phase transition | Alexander Schill |

06.12. | The 1d Ising model, Proof that the critical temperature is >0 in any dimension | Osman Aboubaida |

13.12. | The phase diagram, different characterizations of phase transitions | David Hamann |

2023 |
||

03.01. | - | - |

10.01. | The infinite volume Gibbs state + Correlation inequalities | Arash Roostaei |

17.01. | Peierl's argument: Proof that the critical temperature is finite in d \ge 2 | Matija Blagojevic |

24.01. | Nonzero magnetic field | Romain Akinlami |

31.01. | Infinite volume Gibbs measures: The DLR approach | Jonas Köppl |

07.02. | Crash course on Bernoulli percolation | Xiaohao Ji, Henri Elad Altman |

14.02. | Random-cluster & random-current representations of the Ising model | Wei Huang |

### Literature

Our main reference is the book

- Friedli, Velenik - Statistical Mechanics of Lattice Systems (Cambridge University Press, 2017).