Let \(W = (W(s),\, s\in \mathbb{R}^2_+)\) be a standard Brownian sheet indexed by the nonnegative quadrant. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random open set \(\{(s_1,s_2)\in \mathbb{R}^2_+: W(s_1,s_2) >0\}\) is equal to \(\frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, .\) This result is first established for additive Brownian motion, which provides good local approximations to the Brownian sheet, and then extended, with some technical effort, to the Brownian sheet itself. This is joint work with T. Mountford (Ecole Polytechnique Federale de Lausanne).

A preprint is available at http://arxiv.org/abs/1702.08183.