Hypergeometric SLE and Convergence of Critical Planar Ising Interfaces (Hao Wu, Tsinghua University)

20.11.2017 15:15

Conformal invariance and critical phenomena in two-dimensional statistical physics have been active areas of research in the last few decades. This talk concerns conformally invariant random curves that should describe scaling limits of interfaces in critical lattice models.

The scaling limit of the interface in critical planar lattice model with Doburshin boundary conditions (b.c.), if exists, should satisfy conformal invariance (CI) and domain Markov property (DMP). In 1999, O. Schramm introduced SLE process, and this is the only one-parameter family of random curves with CI and DMP. In 2010, D. Chelkak and S. Smirnov proved that the interface of critical Ising model on the square lattice does converge to SLE(3). In this talk, we discuss the scaling limit of the pair of interfaces in rectangle with alternating b.c. The scaling limit of the pair of interfaces, if exists, should satisfy CI, DMP and symmetry (SYM). It turns out there is a two-parameter family of random curves satisfying CI, DMP, and SYM, and they are Hypergeometric SLE. For the critical Ising model on the square lattice, the pair of interfaces does converge to Hypergeometric SLE(3). In this talk, we will explain two different proofs for the convergence. Furthermore, we will discuss results about global and local multiple SLEs, which correspond to the scaling limit of the collection of interfaces with alternating b.c. in more general setting.

Lieu

Room 17, Séminaire "Mathématique Physique"

entrée libre