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Fluid Interfaces in the 3D Ising Model as a Dilute Gas of Handles

We study the topology of fluid interfaces in the 3D Ising model in the rough phase. It turns out that such interfaces are accurately described as dilute gases of microscopic handles, and the stiffness of the interface increases with the genus. The number of configurations of genus $g$ follows a Poisson-like distribution. The probability per unit area for creating a handle is well fitted in a wide range of the inverse temperature $β$ near the roughening point by an exponentially decreasing function of $β$. The procedure of summing over all topologies results in an effective interface whose squared width scales logarithmically with the lattice size.