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For critical bond-percolation on high-dimensional torus, this paper provessharp lower bounds on the size of the largest cluster, removing a logarithmiccorrection in the lower bound in Heydenreich and van der Hofstad (2007). Thisimprovement finally settles a conjecture by Aizenman (1997) about the role ofboundary conditions in critical high-dimensional percolation, and it is a keystep in deriving further properties of critical percolation on the torus.Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds ondiameter and mixing time of the largest clusters. We further prove that thevolume bounds apply also to any finite number of the largest clusters. The mainconclusion of the paper is that the behavior of critical percolation on thehigh-dimensional torus is the same as for critical Erdos-Renyi random graphs.  In this updated version we incorporate an erratum to be published in aforthcoming issue of Probab. Theory Relat. Fields. This results in amodification of Theorem 1.2 as well as Proposition 3.1.

Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time