For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335-358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551-582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267-1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdós-Rényi random graphs. © 2009 The Author(s).
Heydenreich, M. O., & van der Hofstad, R. (2009). Random graph asymptomics on high-dimensional tori II: volume, diameter and mixing time. Probability Theory and Related Fields. https://doi.org/10.1007/s00440-009-0258-y