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

Source abstract

Let $Z$ be a standard Hermite process of order $m\ge2$ and Hurst index $H\in(1/2,1)$. For $d t\}\sim -\frac12\left(\frac{t}{Λ_m}\right)^{2/m}, t\to\infty,$ 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 $m\to\infty$ with $H$ and $d$ fixed, the normalized constants $\sqrt{m!}\,Λ_m$ 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 $d\le0$, convergence remains uniform up to the memory boundary $H=1/2$. 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 $2^{3/2}π^3G(1/2)^8=2^{11/6}πe\,\mathsf A^{-12}$, where $\mathsf A$ is the Glaisher--Kinkelin constant. The quadratic and cubic coefficients of the fixed-order expansion of the normalized constant as $H\uparrow1$ converge, as $m\to\infty$, to those of the explicit limiting function. Global finite-order bounds and a concentrating-weight limit complement the main result.