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arxiv: 2606.29304 · v1 · pith:HG7H7PNTnew · submitted 2026-06-28 · 🪐 quant-ph · math-ph· math.MP

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

Saikat Sur This is my paper

Pith reviewed 2026-06-30 07:30 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords entanglement entropyErdős–Rényi random graphfree fermionsvolume lawrandom matrix theoryasymptotic freenessthermodynamic limit
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The pith

Ground-state entanglement entropy of free fermions on random graphs follows an exact volume law with a universal density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that the ground-state entanglement entropy for free fermions on an Erdős–Rényi random graph obeys an exact volume law in the thermodynamic limit. The density of entanglement is given by a universal coefficient that does not depend on the probability of edges or other graph details. Numerical checks place this coefficient at about 0.386 nats, below the Page value for random states. The volume law arises because the random graph lacks geometric locality. The proof relies on random matrix theory and asymptotic freeness.

Core claim

The ground-state entanglement entropy obeys an exact volume law in the thermodynamic limit. The entanglement density has a universal coefficient 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.

What carries the argument

Random matrix theory combined with asymptotic freeness applied to the fermionic Hamiltonian on the Erdős–Rényi adjacency matrix, which establishes the volume law for entanglement entropy.

If this is right

  • The entanglement entropy scales linearly with subsystem size instead of its boundary.
  • The universal coefficient remains the same for any edge probability in the random graph.
  • The numerical value of 0.386 nats lies below the Page value expected for typical random states.
  • The result is a direct consequence of the random graph having no underlying geometric structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This suggests that entanglement in mean-field fermionic systems generally follows volume laws rather than area laws.
  • Extensions to other random matrix ensembles or interacting cases could be checked numerically for similar universality.
  • The finding connects to questions about when locality is necessary for area-law entanglement in many-body systems.

Load-bearing premise

Random matrix theory and asymptotic freeness accurately describe the fermionic Hamiltonian on the Erdős–Rényi random graph in the thermodynamic limit.

What would settle it

A calculation or simulation showing that the entanglement density depends on the edge probability for very large graph sizes would falsify the universality claim.

Figures

Figures reproduced from arXiv: 2606.29304 by Saikat Sur.

Figure 1
Figure 1. Figure 1: A schematic representation of the integration contour [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Entropy density s exact as a function of system size N for different values of the edge probability, computed via the exact diagonalization of (1). Each data point represents the average over 100 independent Erdős–Rényi random graph realisations. The entropy density is independent of p for all system sizes considered, consistent with the volume law. (b) The spectral density ρ¯CA (λ) of the entanglement… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to prove, using random matrix theory and asymptotic freeness, that the ground-state entanglement entropy of free fermions on an Erdős–Rényi random graph obeys an exact volume law in the thermodynamic limit, with a universal entanglement density coefficient of approximately 0.386 nats that is independent of edge probability p and microscopic graph details; this is numerically confirmed and attributed to the absence of geometric locality.

Significance. If the result holds, it would establish a rigorous, parameter-free volume law for entanglement entropy in non-local random-graph fermionic systems, with the coefficient strictly below the Page value serving as a falsifiable prediction; the explicit use of asymptotic freeness to derive the universal coefficient and the numerical confirmation are strengths that would advance understanding of entanglement in disordered, non-geometric quantum systems.

major comments (2)
  1. [Abstract] Abstract (and the derivation invoking asymptotic freeness): the central claim of an exact volume law with p-independent coefficient rests on asymptotic freeness applying to the single-particle correlation matrix of the quadratic Hamiltonian H = c† A c with A the symmetric Bernoulli ER adjacency matrix; standard free-probability results require i.i.d. entries of finite variance, but the ER case has dependent entries due to symmetry and the fermionic anticommutators add structure, so explicit verification is needed that no truncation or regularization is required that would affect the entanglement density.
  2. [Numerical confirmation] Numerical section (coefficient confirmation): the value 0.386 is stated as confirmation of the universal coefficient, but without reported details on finite-size scaling, sample averaging, or extrapolation error bars in the N→∞ limit, it is difficult to assess whether the numerics independently support the exact p-independence asserted by the analytic argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the derivation invoking asymptotic freeness): the central claim of an exact volume law with p-independent coefficient rests on asymptotic freeness applying to the single-particle correlation matrix of the quadratic Hamiltonian H = c† A c with A the symmetric Bernoulli ER adjacency matrix; standard free-probability results require i.i.d. entries of finite variance, but the ER case has dependent entries due to symmetry and the fermionic anticommutators add structure, so explicit verification is needed that no truncation or regularization is required that would affect the entanglement density.

    Authors: We appreciate the referee's emphasis on this technical detail. The adjacency matrix A is a symmetric Wigner-type matrix whose upper-triangular entries are independent Bernoulli random variables of finite variance; standard extensions of Voiculescu's asymptotic freeness results to symmetric ensembles with this structure apply directly in the large-N limit. The single-particle correlation matrix is obtained from the spectral decomposition of A, and the fermionic anticommutation relations enter only through the definition of the Gaussian state's entanglement entropy, without introducing additional operator dependencies that violate freeness. We will add an explicit paragraph (in Section II and an appendix) confirming that the requisite conditions hold without truncation or regularization that would alter the entanglement density. revision: yes

  2. Referee: [Numerical confirmation] Numerical section (coefficient confirmation): the value 0.386 is stated as confirmation of the universal coefficient, but without reported details on finite-size scaling, sample averaging, or extrapolation error bars in the N→∞ limit, it is difficult to assess whether the numerics independently support the exact p-independence asserted by the analytic argument.

    Authors: We agree that expanded numerical documentation will make the confirmation more robust. In the revised manuscript we will add a dedicated subsection reporting: the system sizes used for scaling, the number of independent ER realizations averaged for each (p,N) pair, the extrapolation procedure to N→∞ together with statistical error bars, and direct numerical checks of p-independence across several edge probabilities. These additions will allow readers to assess convergence to the reported universal value. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external RMT and freeness

full rationale

The paper states that it proves the exact volume law and p-independent entanglement density using random matrix theory and asymptotic freeness applied to the fermionic Hamiltonian on the Erdős–Rényi adjacency matrix in the thermodynamic limit. The coefficient ~0.386 is described as numerically confirmed rather than fitted or input to the proof. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the derivation chain. The central claim is presented as following from external mathematical results whose hypotheses are taken as given for the model.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract only; full paper unavailable so entries are limited to explicitly invoked tools.

free parameters (1)
  • entanglement density coefficient = 0.386
    Numerically confirmed value stated as approximately 0.386 nats.
axioms (2)
  • domain assumption Random matrix theory applies to the spectrum and eigenvectors of the Erdős–Rényi adjacency matrix
    Invoked to prove the volume law in the thermodynamic limit.
  • domain assumption Asymptotic freeness holds for the relevant operators in the thermodynamic limit
    Central analytical tool for establishing universality and volume law.

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Reference graph

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