Summary: We study the fluctuations of the boundary of a large droplet of one phase of the low-T 2D Ising model in a \(N\times N\) box. It is known that these fluctuations are of the order of \(N^{(1/3)}\) in the vicinity of the walls, and of the order of \(N^{(1/2)}\) far away from the walls. We argue that in fact the fluctuations of this interface can be of any order \(N^b\), \(1/3\Leftarrow b\Leftarrow 1/2\), depending on the location on the interface. We state a conjecture concerning the locations where the fluctuations are of the order \(N^b\).