Finite-Size Scaling and Random Graphs
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We show that the mean-field model for percolation on the high-dimensional torus is the Erdős-Rényi random graph, in the sense that the phase transition for percolation on the high-dimensional torus mimics that of percolation on the complete graph. We first draw inspiration from the Erdős-Rényi random graph. We rigorously prove a number of statements for the Erdős-Rényi random graph, whose proofs can be modified to the setting of high-dimensional tori. We proceed to critical percolation on high-dimensional tori, and extend this discussion to more general tori including the hypercube. We continue by focussing exclusively on the hypercube, where also the supercritical regime is now well understood. Then we discuss scaling limits of critical percolation on random graphs and close this chapter by discussing the role of boundary conditions beyond the periodic boundary conditions that give rise to high-dimensional tori.