Connectivity control of fluctuation strength

Determine whether the correspondence between the free-fermion fluctuation scale and the particle-hole connectivity, expressed asymptotically by N_q approximately equal to z/N, reflects a general connectivity-controlled mechanism for closure beyond the random free-fermion ensemble.

Background

The exact second-moment ratio q_FF in random free fermions is controlled by the determinant depth k and filling rather than directly by the Hilbert-space dimension or the number of channels. At extensive filling, the moment-defined fluctuation scale N_q is asymptotically proportional to the particle-hole connectivity z=k(N-k), normalized by the number of one-body modes.

The paper explicitly cautions that this relation is established as a correspondence within the solvable quadratic model, not as a general control principle. The unresolved problem is to determine whether connectivity generally controls fluctuation strength and the associated closure of the hierarchy in broader models.

References

How this scaling should be organized into an effective connectivity of the microscopic Hamiltonian is left open, and we formulate it explicitly as the sharp question raised by the benchmark.

— Eigenstate thermalization beyond the envelope: exact two-point overlap statistics in random free fermions  (2609.17037 - Huang, 15 Sep 2026) in Section 3, “What controls the fluctuation strength” (Sec. 3.5, around Eq. (qz))

Whether a finite-dimensional vertex ansatz closes generic interacting systems is not claimed by either article and remains the open question.

— Eigenstate thermalization beyond the envelope: exact two-point overlap statistics in random free fermions  (2609.17037 - Huang, 15 Sep 2026) in Section 6, Discussion, paragraph beginning “The exact chain also fixes the form in which a closure of the hierarchy should be sought”