Fluctuations of the Process of Moduli for the Ginibre and Hyperbolic Ensembles

The paper proves the CLT for the hyperbolic ensemble H_α via an exact decomposition of the moduli into independent radii, Riemann-sum asymptotics, and Soshnikov’s theorem, yielding the normalization C(α)_R = α e^R/8 and the variance limit V_f^(α); see Theorem 1.1 and Sections 2.1–2.3 (E S_R and the variance asymptotics) and the statement of Soshnikov’s CLT in the paper (; ; ; ). The candidate solution instead uses the projection kernel identity Var = (1/2) ∬(φ(z)−φ(w))^2|K(z,w)|^2 dμ_α(z)dμ_α(w), performs the angular integration explicitly to a 2F1 hypergeometric term, and extracts the boundary asymptotics—recovering the same centering and variance constant and then invoking Soshnikov’s CLT. This approach aligns with the paper’s general DPP identities (1.2)–(1.3) and its definition of the hyperbolic kernel, albeit with respect to μ_α rather than dm (; ). No substantive conflicts were found; the methods are different but compatible and lead to the same limit law.