Sparse random regular graphs have been proposed as discrete toy models of physical systems with "chaotic" classical dynamics. The paradigm of quantum chaos predicts heuristically that the high energy Laplacian eigenfunctions of such graphs should behave like "random waves" which oscillate rapidly across the edges of the graph --- in particular, deleting all of the edges of the graph where such an eigenfunction changes sign should disconnect the graph into many connected components (which are known as "nodal domains"). We rigorously prove this for eigenfunctions of sufficiently high energy, partially confirming a conjecture made by physicists. The proof employs tools from random matrix theory, graph limits, and combinatorics.
Joint work with Shirshendu Ganguly, Theo McKenzie, and Sidhanth Mohanty.