  We investigate the scaling of the largest critical percolation cluster on a
large d-dimensional torus, for nearest-neighbor percolation in high dimensions,
or when d>6 for sufficient spread-out percolation. We use a relatively simple
coupling argument to show that this largest critical cluster is, with high
probability, bounded above by a large constant times $V^{2/3}$ and below by a
small constant times $V^{2/3}(log V)^{-4/3}$, where V is the volume of the
torus. We also give a simple criterion in terms of the subcritical percolation
two-point function on Z^d under which the lower bound can be improved to small
constant times $V^{2/3}$, i.e., we prove random graph asymptotics for the
largest critical cluster on the high-dimensional torus. This establishes a
conjecture by Aizenman (1997), apart from logarithmic corrections. We discuss
implications of these results on the dependence on boundary conditions for
high-dimensional percolation.
  Our method is crucially based on the results by Borgs, Chayes, van der
Hofstad, Slade and Spencer (2005a, 2005b), where the $V^{2/3}$ scaling was
proved subject to the assumption that a suitably defined critical window
contains the percolation threshold on Z^d. We also strongly rely on mean-field
results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade
(1990) and Hara, van der Hofstad and Slade (2003).
