In recent decades, great progress has been made towards the study of Schramm-Lowener evolution (SLE) utilizing complex analytic and probabilistic techniques. In this talk series, we will mainly discuss current research aspects on multiple SLE, which allows us to introduce notions from conformal field theory (CFT) like fusion and conformal blocks so that we could consider configurations involving many interfaces and construct corresponding partition function for such random curve ensembles. Many progress has been made during recent ten years in that direction, describing novel expression for partition function using PDE, giving hydrodynamical limit, providing more information on describing multiple Ising interfaces, etc. In the end of the seminar, we will focus on a particular model, (radial and chordal) multiple SLE driven by Dyson Brownian motion. We will indicate some relations between the study of such model with random matrix theory. We will also raise some feasible projects to end up the whole series.
This is a paper based reading seminar, which means you need to have certain familarity with some background knowledge on various aspects in geometrical probability (including GFF, SLE, quantum zipper, LQG, percolation, Ising model). We will provide surrogate materials to bridge gaps between your previous knowledge and the topic we want to discuss, however that does not mean you do not require familarity with those basic objects.
The target audience of this seminar are those students who have certain exposure to the current research on the side of geometrical probability.
[KP16] K. Kytola and E. Peltola. Pure partition functions of multiple SLEs. Commun. Math. Phys., 346:237–292, 2016.
[Dub09] J. Dubedat. SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc., 22:995–1054, 2009.
[BPW18] V. Beffara, E. Peltola, and H. Wu. On the uniqueness of global multiple SLE, 2018. arXiv:1801.07699.
[Car03a] J. Cardy. Stochastic Loewner evolution and Dyson’s circular ensembles. J. Phys. A: Math. Gen., 36:L379–L386, 2003.
[Car03b] J. Cardy. Corrigendum: Stochastic Loewner evolution and Dyson’s circular ensembles. J.Phys. A: Math. Gen., 36:12343, 2003.
[HK18] I. Hotta and M. Katori. Hydrodynamic limit of multiple SLE. J. Stat. Phys., 171:166–188, 2018.
[Izy20] K. Izyurov. On multiple SLE for the FK-Ising model, 2020. arXiv:2003.08735.
[Kar19] A. Karrila. Multiple SLE type scaling limits: from local to global, 2019. arXiv:1903.10354
[Kos19] S. Koshida. Multiple backward Schramm–Loewner evolution and coupling with Gaussian free field, 2019. arXiv:1908.07180.
[Pel19] E.Peltola (2019). Toward a conformal field theory for Schramm-Loewner evolutions. Journal of Mathematical Physics , 60 (10), 103305.