This talk will be about random lozenge tilings of a class of planar domains which I like to call "sawtooth domains." The basic question is: what does a uniformly random tiling of a large sawtooth domain look like? At the first order of randomness, a remarkable form of the law of large numbers emerges: the height function of the tiling converges to a deterministic "limit shape." My talk is about the next order of randomness, where one wants to analyze the fluctuations of tiles around their eventual positions in the limit shape. Quite remarkably, this ostensibly analytic problem can be solved in an essentially combinatorial way, using a desymmetrized version of the double Hurwitz numbers from enumerative algebraic geometry.