  Using a large database of 8 million institutional trades executed in the U.S.
equity market, we establish a clear crossover between a linear market impact
regime and a square-root regime as a function of the volume of the order. Our
empirical results are remarkably well explained by a recently proposed
dynamical theory of liquidity that makes specific predictions about the scaling
function describing this crossover. Allowing at least two characteristic time
scales for the liquidity (`fast' and `slow') enables one to reach quantitative
agreement with the data.
