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

For critical (bond-) percolation on general high-dimensional torus, this paper answers the following questions: What is the diameter of the largest cluster? What is the mixing time of simple random walk on the largest cluster? The answer is the same as for critical Erd¿os-R´enyi random graphs, and extends earlier results by Nachmias and Peres [35] in this setting. We further improve our bound on the size of the largest cluster in [24], and extend the results on the largest clusters in [9, 10] 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. This result further justifies that the critical value defined in [9, 10] is appropriate in our rather general setting of random subgraphs of high-dimensional tori.