  We study the possible scaling limits of percolation interfaces in two
dimensions on the triangular lattice. When one lets the percolation parameter
p(N) vary with the size N of the box that one is considering, three
possibilities arise in the large-scale limit. It is known that when p(N) does
not converge to 1/2 fast enough, then the scaling limits are degenerate,
whereas if p(N) - 1 / 2 goes to zero quickly, the scaling limits are SLE(6) as
when p=1/2. We study some properties of the (non-void) intermediate regime
where the large scale behavior is neither SLE(6) nor degenerate. We prove that
in this case, the law of any scaling limit is singular with respect to that of
SLE(6), even if it is still supported on the set of curves with Hausdorff
dimension equal to 7/4.
