Supersymmetry is a powerful tool to study problems in probability and mathematical physics such as random walks with and without memory, random matrices and quantum chaos. It allows to prove some deep results that are inaccessible to other techniques. One example is the proof of the long standing open conjecture about existence of a phase transition for a random walk with memory in dimension 3 or higher. The basic idea is to represent certain probabilistic averages over many interacting degrees of freedom in terms of a generalised statistical mechanics model in significantly lower dimensions, thus simplifying the problem drastically. The emerging model consists of interacting particles with generalised spins that are represented by vectors/matrices whose elements are real (commuting) or 'Grassmann' (anticommuting) variables. The goal of this seminar is to give an introduction to the supersymmetric approach and provide some applications. We will start by introducing all the necessary tools, including Grassmann variables and supersymmetric integrals. Depending on the interest of the participants we will continue studying some examples in random matrices and/or random walks. A list of possible topics can be found here.
If you want to participate in the seminar, please send an email to torben.krueger(at)uni-bonn.de. In case you would like to give a talk and already have a preference, inform me about your preferred topic and/or about your background (so that I can assign an appropriate one). There are still slots available! If you need a certificate you also have to register in BASIS.
The seminar takes place on Fridays at 12PM (c.t.) in SR N0.003. The following preliminary schedule may still change until the beginning of the semester.
Date
April 05
April 12
April 19
April 26
May 03
May 10
May 17
May 24
May 31
June 07
June 14
Topic
Introduction to Grassmann variables
No seminar
No seminar
The matrix tree theorem
No seminar
Classical BFS-Dynkin isomorphism
Supersymmetric BFS-Dynkin isomorphism
Mermin-Wagner theorem for vertex reinforced jump process
Density of states for the Gaussian unitary ensemble
Anderson localization at weak disorder
No seminar
The participants are expected to have a basic knowledge of linear algebra, functional analysis, probability theory.
Every participant who needs a certificate is expected to prepare a talk of approximately 90 minutes (projector or blackboard) that reflects a good understanding of the underlying literature source and a written summary of the chosen topic that is to be provided to the audience. The talks can be based on chapters of books or research articles. The reference list below provides ample examples but based on the preferences of the participants many other topics are possible.
[1] A. Abdesselam. The grassmann-berezin calculus and theorems of the matrix-tree type. Advances in Applied Mathematics, 33(1):51 – 70, 2004.
[2] R. Bauerschmidt. Supersymmetry for probabilists. http://www.statslab.cam.ac.uk/ rb812/teaching/toronto2018/toronto.pdf, 2018.
[3] R. Bauerschmidt, T. Helmut, and A. Swan. Dynkin isomorphism and mermin–wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process. arXiv:1802.02077, 2018.
[4] D. C. Brydges, J. Z. Imbrie, and G. Slade. Functional integral representations for self-avoiding walk. Probab. Surveys, 6:34–61, 2009.
[5] J. Bunder, K. B. Efetov, V. E. Kravtsov, O. M. Yevtushenko, and M. R. Zirnbauer. Superbosonization formula and its application to random matrix theory. Journal of Statistical Physics, 129, 07 2007.
[6] M. Disertori. Density of states for gue through supersymmetric approach. Reviews in Mathematical Physics - RMP, 16:1191–1225, 10 2004.
[7] M. Disertori and M. Lager. Density of states for random band matrices in two dimensions. Annales Henri Poincaré, 18(7):2367–2413, Jul 2017.
[8] M. Disertori, C. Sabot, and P. Tarrès. Transience of edge-reinforced random walk. Communications in Mathematical Physics, 339(1):121–148, Oct 2015.
[9] K. Efetov. Supersymmetry in Disorder and Chaos. Cambridge University Press, 1996.
[10] C. Sabot and P. Tarrès. Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model. Journal of the European Mathematical Society, 2015. 18 pages.