| Dr. Lorenzo Taggi and Dr. Benedikt Jahnel |

| Lorenzo.Taggi@wias-berlin.de and Benedikt.Jahnel@wias-berlin.de |

| Room 413, Mohrenstraße 39, 10117 Berlin |

Lecture

In the lecture, in the first part of the semester, we present the general probabilistic framework of statistical mechanics in infinite volume. That is, we consider systems of infinite configurations of components in space that carry some random local information, their spins. The probability to see a configuration then depends on the mutual energy of the system, introducing dependencies. Apart from its original intention to describe phenomena in physics, like spontaneous magnetization and other macroscopic phase-transition phenomena, the approach of statistical mechanics has found numerous applications in various parts of applied mathematics, for example probabilistic biology, stochastic geometry or even data science. We will focus on the essential parts of the general theory, e.g., existence and uniqueness results, which we illustrate via paradigmatic models such as the famous Ising model and more general spin systems with continuous symmetries.
As a prerequisite for the course we need a solid foundation in measure-theoretic probability. The lecture will be given as a 4SWS lecture in the first half of the summer semester 2020. In the second half of the semester we offer a seminar on the same topic, for which the lecture provides the necessary background.

Seminar

In the seminar, in the second part of the semester, we present and discuss topics of statistical mechanics in infinite volume, more precisely, spin systems and associated macroscopic phase transitions. Statistical mechanics, apart from its original intention to describe phenomena in physics, like spontaneous magnetization and other phase-transition phenomena, has found numerous applications in various parts of applied mathematics, for example probabilistic biology, stochastic geometry or even data science. This seminar builds on the 2SWS lecture with the same title that will be given by the instructors in the first half of the summer semester 2020. The individual topics will be distributed in early May 2020 and presentations should be delivered in the second half of the semester.

Mode of Operation and Dates

The original plan was to have a classical lecture and seminar on the following dates:
The first video will be made available on Tuesday 21 April 2020.

The lecture phase is from 21 April 2020 until 05 June 2020.

The seminar phase is from 09 June 2020 until approximately 21 July 2020.

Videos

- 21 April 2020:
- 1. Introduction - spin systems
- 2. Introduction - phase transitions
- 3. Ising model - definition
- 4. Ising model - high / low temperatur
- Slides from the lecture

- 24 April 2020:
- 5. Ising model - pressure
- 6. Ising model - magnetization
- 7. Ising model - one dimension
- Slides from the lecture
- Comments and corrections:
- Video 5: The parallelpipeds Lambda and Delta have to be disjoint; misspelling of edge set at 18:36 - B^b must be E^b
- Video 6: Misspelling in the corollary about properties of the magentization item 1): It holds for all h not in B_beta
- Video 7: When I write "Laplace principle", I here mean the elementary property that can be found for example in Lemma 1.2.15 (Large Deviations Techniques and Applications - Dembo / Zeitouni Springer '98)

- 28 April 2020:
- 8. Ising model - Gibbs states
- 9. Ising model - existence via correllation inequalities
- Slides from the lecture
- Comments and corrections:
- Video 9: The alternating sequence around 31:20 is a subsequence of Lambda^1 on uneven indices and a subsequence of Lambda^2 on even indices. Since Lambda^1 and Lambda^2 are cofinal, these subsequences can be chosen such the resulting sequence is cofinal as well.

- 05 May 2020:
- 10. Ising model - phase diagram and characterizations
- 11. Ising model - uniqueness in the symmetric model
- Slides from the lecture
- Comments and corrections:
- Video 11: In the last inequalities, E_o is also connected and contains the origin.

- 12 May 2020:
- 12. Ising model - non-uniqueness in the high-temperature symmetric model
- 13. Ising model - Peierls' argument
- Slides from the lecture

- 15 May 2020:
- 14. Ising model - uniqueness in the asymmetric model
- 15. Ising model - Lee-Yang circle theorem
- Slides from the lecture

- 19 May 2020:
- 16. Welcome 2
- 17. Ising model - correlations - goal and overview
- 18. Ising model - correlations - Ising model on graphs
- 19. Ising model - correlations - small correction
- 20. Ising model - correlations - definition of random currents
- 21. Ising model - correlations - from currents to spins
- 22. Ising model - correlations - proof of main theorem (part 1)
- 23. Ising model - correlations - proof of main theorem (part 2)
- Slides from the lecture

- 22 May 2020:
- 24. Ising model - correlations - proof of insertion tolerance lemma
- 25. Ising model - correlations - proof of switching lemma
- Slides from the lecture

- 26 May 2020:
- 26. GFF - Introduction and overview
- 27. GFF - Gaussian Vectors
- 28. GFF - Discrete Green Identities
- 29. GFF - The height function is a gaussian vector: first step
- Slides from the lecture

- 29 May 2020:
- 30. GFF - Dirichlet problem
- 31. GFF - The massless case: covariance matrix
- 32. GFF - Thermodynamic limit and (non) uniqueness in the massless case
- Slides from the lecture

- 02 June 2020:
- 33. Spin systems with continuous symmetry - definitions and goals
- 34. Spin systems with continuous symmetry - heuristic argument
- 35. Spin systems with continuous symmetry - proof of the Mermin-Wagner theorem
- Slides from the lecture

- 05 June 2020 (last lecture):
- 36. Spin systems with continuous symmetry - end of the proof and minimisation of the Dirichlet energy
- 37. Spin systems with continuous symmetry - decay of correlations in the 2d XY model
- 38. Spin systems with continuous symmetry - characterisation of the maximiser
- 39. Spin systems with continuous symmetry - concluding remarks
- Slides from the lecture

References

The lecture will be based on the following book:
We also recommend these books on the topic, which are also available as ebooks:

Those three books are also blocked in the mathematical library of the TU Berlin for our course, see primo search portal