Volume Law and Universality of Entanglement Entropy in Random Graph Fermi Systems

We study the ground-state entanglement entropy of free fermions on the Erd\H{o}s--R\'enyi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we prove that the ground-state entanglement entropy obeys an exact volume law in the thermodynamic limit. The entanglement density, with a universal coefficient that is independent of the edge probability and the microscopic details of the graph. This coefficient is confirmed numerically to take the value approximately $0.386$ nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.