Uniform sine-kernel determinant asymptotics, tail-side quantiles, and prolate eigenvalue bounds

Abstract

Let $S_c=P_{(0,c)}Q{P}_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#n:{\lambda }_n(c)>a$, and set $\bar{L}=\log ((1-\delta )/\delta )$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_{\delta }(c)=c+{\pi }^{-2}\bar{L}\log (4{\pi }^{2}c/\bar{L})+O_A(\log c+\bar{L})$ for $6\le \bar{L}\le A\log c$. It yields corresponding additive formulas for the lower half and full plunge, with main terms respectively ${\pi }^{-2}\bar{L}\log (4{\pi }^{2}c/\bar{L})$ and twice this quantity. An exact one-tail-coordinate selection gives, for fixed $A>0$, $d\ge 1$, and $q\in (1/2,1)$, the one-sided tensor-product bound ${\Lambda }_{\delta }(c;d)\ge {\pi }^{-2}d{c}^{d-1}\bar{L}\log (4{\pi }^{2}c/\bar{L})-O_{A,d,q}(c^{d-1}(\log c+\bar{L}))$ for $L_{d,q}\le \bar{L}\le A\log c$, where $L_{d,q}=\log (q^{-(d-1)}(e^6+1)-1)$; the tensor content is nontrivial for $d\ge 2$. The analytic input is a signed growing-parameter sine-kernel determinant asymptotic: uniformly for $0\le \omega \le A\log s$, $\log \det (I+(e^{2\omega }-1)K_s)=4\omega s/\pi +2{\pi }^{-2}{\omega }^2\log (4s)+2\log |G(1+i\omega /\pi ){|}^2+O_A((1+\omega {)}^4{\log }^{2}s/s)$, where $G$ is the Barnes $G$-function. We prove this negative-coupling counterpart of the Bothner--Deift--Its--Krasovsky theorem by direct IIKS steepest descent. We also retain the uniform head-side results and use a two-way determinant reduction to obtain the moving-depth lower-half bridge bound with constant $1/(32{\pi }^2)$; extending it to the deeper range uses Kulikov--Dam Larsen and may require a smaller constant. These counting formulas are additive. Their errors become uniformly relative when $\bar{L}$ tends uniformly to infinity; fixed thresholds are covered separately by Landau--Widom. A Lambert-$W_{-1}$ formula is recorded only for the continuous main term, not for individual eigenvalues.

69 pages, no figures. Revised and substantially expanded version. Adds tail-side quantile results and their lower-half, full-plunge, and tensor-product consequences; expands the signed sine-kernel determinant analysis; and clarifies the uniformity ranges and references. The main conclusions of the previous version are retained