Speaker
Kalle Kytölä
(University of Helsinki)
Description
We consider the square lattice Ising model at its critical
point in simply connected domains with a
boundary. We split the boundary to a few pieces, and impose
different boundary conditions on each. We are
interested in the scaling limit in which a given domain is
approximated by subgraphs of the square lattice
with mesh size tending to zero. In the scaling limit
conformal invariance properties are expected if the
boundary conditions are combinations of plus, minus and
free. A traditional interpretation of conformal
invariance concerns correlation functions, and our first
results are explicit expressions for some
correlation functions of boundary spins in terms of the
Riemann uniformizing map of the domain.
A different point of view to conformal invariance, initiated by Schramm, is to focus attention to random curves or interfaces. In the Ising model, Smirnov's work shows conformal invariance of two kinds of interfaces: an exploration path in the FK representation of Ising with plus-free boundary conditions tends to the chordal SLE(16/3) process, and a curve in the low temperature expansion with plus-minus boundary conditions tends to the chordal SLE(3) process. Our work uses Smirnov's first result to obtain a generalization of the second. The generalization concerns a curve in the low temperature expansion with free-plus-minus boundary conditions. We will show how the expressions for correlation functions identify the limit of this curve as a variant of SLE(3) called the dipolar SLE(3). This generalization was first conjectured by Bauer & Bernard & Houdayer.
The presentation is based on joint work with Clément Hongler (Université de Genève & Columbia University).
A different point of view to conformal invariance, initiated by Schramm, is to focus attention to random curves or interfaces. In the Ising model, Smirnov's work shows conformal invariance of two kinds of interfaces: an exploration path in the FK representation of Ising with plus-free boundary conditions tends to the chordal SLE(16/3) process, and a curve in the low temperature expansion with plus-minus boundary conditions tends to the chordal SLE(3) process. Our work uses Smirnov's first result to obtain a generalization of the second. The generalization concerns a curve in the low temperature expansion with free-plus-minus boundary conditions. We will show how the expressions for correlation functions identify the limit of this curve as a variant of SLE(3) called the dipolar SLE(3). This generalization was first conjectured by Bauer & Bernard & Houdayer.
The presentation is based on joint work with Clément Hongler (Université de Genève & Columbia University).
