Cycle structure of percolation on high-dimensional tori

In the past years, many properties of the critical behavior of the largest connected components 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 Erdős-Rényi random graph, raising the question whether the scaling limits or 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 highdimensional percolation on the torus {−br/2c, . . . , dr/2e − 1}. 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, as on the critical Erdős-Rényi random graph. On the Erdős-Rényi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by Addario-Berry, Broutin and Goldschmidt (2010). Our proofs crucially rely on various new estimates of probabilities of the existence of open paths in critical Bernoulli percolation on Z with constraints on their lengths. We believe these estimates are interesting in their own right.