The Critical Window for High-Dimensional Limits of Hyperbolic Poisson k-Plane Processes

For 2k > d + 1, the centred and normalized total k-volume of a stationary Poisson process of k-planes in hyperbolic space H^d, observed in a growing ball, converges to a non-Gaussian infinitely divisible law Z_{d,k}. Bühler and Hug asked how the standardized law Z*_{d,k} = Z_{d,k}/(Var Z_{d,k})^{1/2} behaves as d → ∞. Bühler, Hug and Thäle proved that, when k/d → 1/2, Z*_{d,k} tends to the standard Gaussian law if d^{−1}(2k − d − 1)^{d/k} stays asymptotically below eπ and to 0 if it stays above eπ, and they left the critical rate open. We complete the picture. Put m = 2k − d − 1, m_c = (2πe(k − 1))^{1/2} and y = (m − m_c)/m_c^{1/2}. Along every sequence of admissible pairs with d → ∞, the Lévy distance between Z*_{d,k} and the centred Gaussian law with variance Φ(−y/√2) tends to 0, where Φ is the standard normal distribution function. Hence Z*_{d,k} converges in distribution if and only if y converges in [−∞, ∞], every limit law is a centred, possibly degenerate, Gaussian N(0, τ²) with τ² ∈ [0, 1] (where N(0, 0) = δ₀), and every τ² ∈ [0, 1] occurs. At the critical rate k = d/2 + ½(eπd)^{1/2} + O(1) the limit is N(0, 1/2); a shift by c·d^{1/4} gives N(0, Φ(−√2·c/(eπ)^{1/4})). The proof combines an exact Beta representation of the Kolmogorov measure, a reduction lemma for Kolmogorov measures that split between 0 and ∞, an anti-concentration bound, and a central limit theorem for the logarithm of a Beta variable. This is an unrefereed note.