Title: Critical 2d Ising model with mixed boundary conditions
Abstract:
We consider the square lattice Ising model at its critical point in
simply connected domains with a boundary. The boundary is split to a
few pieces, different boundary conditions are imposed 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 one expects conformal invariance
properties if the chosen boundary conditions are combinations of plus,
minus and free. Our first results are explicit, conformally covariant
expressions for some correlation functions.
In an approach pioneered by Lawler & Schramm & Werner, conformal
invariance is addressed using random geometry, focusing attention to
curves or interfaces in the model. For the Ising model, Smirnov has
shown 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, first conjectured by Bauer &
Bernard & Houdayer, is a crucial first step towards a random geometry
description of the full scaling limit of the Ising spin clusters with
free boundary conditions.
The talk is based on joint work with Cl?ment Hongler (Universit? de
Gen?ve and Columbia University).