A Gaussian cousin to the Erdős–Rényi graph
The discrete Gaussian Free Field (GFF) yields a percolation model on transient graphs that has been subject to much attention, in particular on the lattice Z^d (for fixed d>2). One can extend its definition to a typical finite d-regular graph. When its number of vertices becomes large, we study the level-set above h, i.e. the subgraph formed by vertices whose GFF value is larger than some real parameter h. If h is below a certain critical threshold h*, we establish the emergence of a single huge component containing a positive proportion of the vertices, sharing many similarities with the giant component of the classical Erdős-Rényi model.
The argument relies on a comparison with the GFF on the d-regular tree, which yields an infinite-type branching process. We also show that in the supercritical case h<h*, the random walk on the level-set of the tree has a positive speed.