We study the fluctuations of the boundary of a large droplet of one phase of the low-T 2D Ising model in a N x 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 <= b <= 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.
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Ott et al. (2024) studied this question.
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