Discrete harmonic functions in the quadrant

In this talk we shall be interested in discrete harmonic functions in cones (in particular, in the quarter plane). The generating function of these harmonic functions satisfies a functional equation (closed to a well-known functional equation that appears in the context of enumeration of confined walks in combinatorics). We shall show the link between these harmonic functions and a one-parameter family of conformal mappings. One of the motivations to that study is to condition (in the sense of Doob) random walks never to leave cones.