  In the past years, many properties of the largest connected components of
critical percolation on the high-dimensional torus, such as their sizes and
diameter, have been established. The order of magnitude of these quantities
equals the one for percolation on the complete graph or Erdos-Renyi random
graph, raising the question whether the scaling limits of the largest connected
components, as identified by Aldous (1997), are also equal.
  In this paper, we investigate the cycle structure of the largest critical
components for high-dimensional percolation on the torus (Z/rZ)^d. While
percolation clusters naturally have many short cycles, we show that the long
cycles, i.e., cycles that pass through the boundary of the cube of width r/4
centered around each of their vertices, have length of order r^{d/3}, as on the
critical Erdos-Renyi random graph. On the Erdos-Renyi random graph, cycles play
an essential role in the scaling limit of the large critical clusters, as
identified by Addario-Berry, Broutin and Goldschmidt (arXiv:0908.3629).
  Our proofs crucially rely on various new estimates of probabilities of the
existence of open paths in critical Bernoulli percolation on Z^d with
constraints on their lengths. We believe these estimates are interesting in
their own right.
