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).