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arXiv:2606.29304

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

Saikat Sur

quant-phmath-phSubmitted Jun 28, 2026• Announced Jun 30, 2026• 0 citations
v1 · Jun 28, 2026v2 · Jul 6, 2026
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Abstract

We study the ground-state entanglement entropy of free fermions on the Erdős--Rényi 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.

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We study the ground-state entanglement entropy of free fermions on the Erdős--Rényi 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 0.3860.3860.386 nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.
We consider a quantum system of large size N and its subsystem of size L, assuming that N is much larger than L, which can also be sufficiently large, i.e., 1≪L≲N. A widely accepted mathematical version of this inequality is the asymptotic regime of successive limits: first the macroscopic limit N→∞, then an asymptotic analysis of the entanglement entropy as L→∞. In this paper, we consider another version of the above inequality: the regime of asymptotically proportional L and N, i.e., the simultaneous limits L→∞,N→∞,L/N→λ>0. Specifically, we consider a system of free fermions that is in its ground state, and such that its one-body Hamiltonian is a large random matrix, which is often used to model long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give streamlined proof of Page’s formula for the entanglement entropy of black hole radiation for a wide class of typical ground states, thereby proving the universality and the typicality of the formula.
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We investigate multipartite information and entanglement measures in the ground state of a free-fermion model defined on a Hamming graph. Using the known diagonalization of the adjacency matrix, we solve the model and construct the ground-state correlation matrix. Moreover, we find all the eigenvalues of the chopped correlation matrix when the subsystem consists of n disjoint Hamming subgraphs embedded in a larger one. These results allow us to find an exact formula for the entanglement entropy of disjoint graphs, as well as for the mutual and tripartite information. We use the exact formulas for these measures to extract their asymptotic behavior in two distinct thermodynamic limits, and find excellent match with the numerical calculations. In particular, we find that the entanglement entropy admits a logarithmic violation of the area law which decreases the amount of entanglement compared to the area law scaling.