From Hermite Chaos to an Attractive Log-Gas: Explicit High-Rank Limits of Iterated-Logarithm and Tail Constants

@misc{moldavskaya2026from,
  title = {{From Hermite Chaos to an Attractive Log-Gas: Explicit High-Rank Limits of Iterated-Logarithm and Tail Constants}},
  author = {Elina Moldavskaya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27261},
  eprint = {2610.27261},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27261},
}

TY  - UNPB
TI  - From Hermite Chaos to an Attractive Log-Gas: Explicit High-Rank Limits of Iterated-Logarithm and Tail Constants
AU  - Elina Moldavskaya
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27261
UR  - https://arxiv.org/abs/2610.27261
AB  - Let Z be a standard Hermite process of order m≥2 and Hurst index H∈(1/2,1). For d<H−1/2, let Y be the variance-normalized version of the weighted integral ∫_0^1 x^−d dZ(x). Its right tail satisfies $ \log\Pr\{Y>t\}\sim -\frac12\left(\frac{t}{Λ_m}\right)^{2/m}, t\to\infty, $ where the positive constant Λ_m admits an infinite-dimensional variational representation. The same constant governs the sharp law of the iterated logarithm for weighted sums of nonlinear functions of long-memory Gaussian sequences. A general closed-form evaluation at finite order is not known. As m→∞ with H and d fixed, the normalized constants √(m!) Λ_m decrease to an explicit positive function of H and d, with relative error O(m^−1). This function also gives a lower bound for the normalized constant at every finite order. An exact Rényi variational representation at each order yields, in the high-rank limit, the entropy--energy variational problem of an attractive logarithmic gas. The limiting free energy is evaluated through the Selberg integral and the Barnes G function. For d≤0, convergence remains uniform up to the memory boundary H=1/2. After division by the square-root boundary factor √(2H−1), the high-rank and boundary limits commute. Their common value is explicit; for constant weights it equals 2^3/2π^3G(1/2)^8=2^11/6πe A^−12, where A is the Glaisher--Kinkelin constant. The quadratic and cubic coefficients of the fixed-order expansion of the normalized constant as H↑1 converge, as m→∞, to those of the explicit limiting function. Global finite-order bounds and a concentrating-weight limit complement the main result.
ER  -