UNIVERSAL EIGENVALUE STATISTICS FOR DYNAMICALLY DEFINED MATRICES

The paper rigorously proves a local law and bulk universality for the Hermitized matrix HX built from a single doubling-map orbit, including the precise self-consistent equation ∫_0^1 1/(g_f(x) m∞(z) + z) dx = − m∞(z) and the GUE test-function comparison at the 1/(2N ρ∞(E)) scale (Theorems 2.5 and 2.9) . A key technical insight of the paper is that existing correlated-matrix frameworks do not apply directly to this model due to its complete deterministic dependence: even correlations of bounded functions of entries need not decay with distance (their condition (2.15) fails) . The authors therefore introduce a novel resampling that yields only logarithmic range of dependence (HY) and then prove local law and universality for HY and transfer the results back to HX via Green function comparison and a DBM step, while deriving and justifying the scalar self-consistent equation via a Toeplitz-symbol analysis of gf . By contrast, the model’s solution asserts that HX already lies in an AEK-type short-range class and that one may directly invoke the general MDE local law and DBM universality. This overlooks the paper’s explicit obstacle (non-decaying correlations of bounded functions) and does not rigorously verify the cumulant/mixing hypotheses required for the AEK theory in this non-Gaussian, single-orbit setting. While the model correctly reproduces the limiting scalar equation and the claimed universality statement, its core justification for applying AEK directly to HX is unsupported and contradicts the paper’s analysis; thus, as presented, the model’s proof is not valid for this ensemble.