Flocks of birds exhibit a remarkable degree of coordination and collective
response. It is not just that thousands of individuals fly, on average, in the
same direction and at the same speed, but that even the fluctuations around the
mean velocity are correlated over long distances. Quantitative measurements on
flocks of starlings, in particular, show that these fluctuations are
scale-free, with effective correlation lengths proportional to the linear size
of the flock. Here we construct models for the joint distribution of velocities
in the flock that reproduce the observed local correlations between individuals
and their neighbors, as well as the variance of flight speeds across
individuals, but otherwise have as little structure as possible. These
minimally structured, or maximum entropy models provide quantitative,
parameter-free predictions for the spread of correlations throughout the flock,
and these are in excellent agreement with the data. These models are
mathematically equivalent to statistical physics models for ordering in
magnets, and the correct prediction of scale-free correlations arises because
the parameters - completely determined by the data - are in the critical
regime. In biological terms, criticality allows the flock to achieve maximal
correlation across long distances with limited speed fluctuations
