  We consider two-dimensional critical bond percolation. Conditioned on the
existence of an open circuit in an annulus, we show that the ratio of the
expected size of the shortest open circuit to the expected size of the
innermost circuit tends to zero as the side length of the annulus tends to
infinity, the aspect ratio remaining fixed. The same proof yields a similar
result for the lowest open crossing of a rectangle. In this last case, we
answer a question of Kesten and Zhang by showing in addition that the ratio of
the length of the shortest crossing to the length of the lowest tends to zero
in probability. This suggests that the chemical distance in critical
percolation is given by an exponent strictly smaller than that of the lowest
path.
