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

Let $Z$ be a standard Hermite process of order $m\ge 2$ and Hurst index $H\in (1\mathrm{/}2,1)$. For $d<H-1\mathrm{/}2$, let $Y$ be the variance-normalized version of the weighted integral ${\int }_0^1{x}^{-d}\mathrm{d}Z(x)$. Its right tail satisfies $\log \mathrm{Pr}\{Y>t\}\sim -\frac{1}{2}{\left(\frac{t}{{\mathrm{\Lambda }}_m}\right)}^{2\mathrm{/}m},t\to \mathrm{\infty },$ where the positive constant ${\mathrm{\Lambda }}_m$ 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 $m\to \mathrm{\infty }$ with $H$ and $d$ fixed, the normalized constants $\sqrt{m!}{\mathrm{\Lambda }}_m$ 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 $d\le 0$, convergence remains uniform up to the memory boundary $H=1\mathrm{/}2$. After division by the square-root boundary factor $\sqrt{2H-1}$, the high-rank and boundary limits commute. Their common value is explicit; for constant weights it equals $2^{3\mathrm{/}2}{\pi }^{3}G(1\mathrm{/}2{)}^8=2^{11\mathrm{/}6}\pi e{\mathsf{A}}^{-12}$, where $\mathsf{A}$ is the Glaisher–Kinkelin constant. The quadratic and cubic coefficients of the fixed-order expansion of the normalized constant as $H\uparrow 1$ converge, as $m\to \mathrm{\infty }$, to those of the explicit limiting function. Global finite-order bounds and a concentrating-weight limit complement the main result.