Sharp Variance-Aware Bennett and Bernstein Concentration Inequalities for General Linear Statistics of Determinantal Point Processes

We establish a unified family of sharp, non-asymptotic variance-aware Bennett and Bernstein concentration inequalities with explicit optimal constants for general linear statistics S_w = ∑ᵢ₌₁^n w_i X_i of discrete and continuous self-adjoint Determinantal Point Processes (DPPs) equipped with arbitrary real weights w ∈ ℝ^n. Classical concentration bounds under Negative Association (Joag-Dev and Proschan, 1983) suffer from the independent variance deficit: moment generating function domination bounds the joint Laplace transform by an independent product measure, forcing the variance proxy in the exponential tail bound to be the crude sum of marginal variances ∑ᵢ₌₁^n w_i^2 Var(X_i). This forfeit of the strictly negative cross-covariances −∑i≠j w_i w_j Kᵢⱼ^2 induced by fermionic repulsion is severe in repulsive physical lattices, volume sampling, and projection DPPs, where the true ensemble variance σ_w^2 = Var(S_w) is orders of magnitude smaller than the independent proxy, or vanishes identically. Conversely, modified log-Sobolev and Lipschitz concentration bounds for Strongly Rayleigh measures (Pemantle and Peres, 2014; Hermon and Salez, 2023) are Hoeffding-type bounds governed by the ambient sample size or dimension, remaining blind to σ_w^2, whereas the spectral Bernoulli representation of point counts (Hough et al., 2006) collapses for non-constant weights because W = diag(w) and K do not commute (WK ≠ KW). While Fredholm-determinant cumulant expansions yield variance-aware sub-exponential concentration for projection DPPs (Breuer and Duits, 2014) and general Hermitian or non-symmetric DPPs (Bardenet et al., 2024), power-series majorization introduces unspecified or non-optimal universal constants in the exponent, lacks the exact Bennett rate function (failing to annihilate at zero variance), and treats one-sided and signed weights identically. We resolve these sharp-constant and rate-function barriers through an exact operator-theoretic and dynamical framework: (i) Continuous Matrix Riccati Flow: Under exponential tilting dℙ_λ ∝ eλ S_w dℙ, the tilted marginal kernel K_λ satisfies the autonomous matrix Riccati differential equation dK_λ / dλ = (1/2){W, K_λ} − K_λ W K_λ, holding unconditionally for all 0 ≼ K ≼ I. (ii) Intertwining Operator Symmetry and Spectral Invertibility: The matrix pencil F(λ) = I + (D_λ − I)K is cospectral with the strictly positive definite symmetric operator S(λ) = (I − K) + K1/2 D_λ K1/2 ≻ 0, with explicit uniform spectral lower bound λ_min(F(λ)) = λ_min(S(λ)) ≥ exp(−|λ| ‖w‖_∞) > 0. We establish the intertwining operator identity S(λ) K1/2 = K1/2 F(λ), providing the manifest symmetric Gram factorization K_λ = (K1/2 D_λ1/2)^T S(λ)⁻¹ (K1/2 D_λ1/2) and proving unconditionally 0 ≼ K_λ ≼ I without requiring strict contractivity K ≺ I. (iii) Exact Isometry Block Embedding in ℝ²ⁿ: By embedding ℝ^n into the doubled space ℝ²ⁿ via the isometry U_λ = [K_λ1/2; (I − K_λ)1/2], we uncover an exact operator representation for the tilted third cumulant V'(λ) = ψ_w'''(λ) = Tr(B_λ^T C_λ B_λ) − Tr(B_λ A_λ B_λ^T), where B_λ = (I − K_λ)1/2 W K_λ1/2, A_λ = K_λ1/2 W K_λ1/2, and C_λ = (I − K_λ)1/2 W (I − K_λ)1/2. (iv) Universal Operator Differential Inequality and Sharpness: We prove the master differential relation |V'(λ)| ≤ c_w ‖w‖_∞ V(λ), where c_w = 1 for one-sided (non-negative or non-positive) weights, and c_w = 2 for arbitrary signed weights w ∈ ℝ^n. Using a rank-1 projection extremal family, we show explicitly that |V'(0)| / (‖w‖_∞ V(0)) = 2 − 4p → 2 as p → 0^+, proving that the constant c_w = 2 is unimprovable. (v) Sharp Bennett, Bernstein, and Moment Bounds: Grönwall integration and convex Fenchel–Legendre inversion yield tail bounds driven strictly by the exact ensemble variance σ_w^2 = ‖(I − K)1/2 W K1/2‖_F^2: an exact Bennett inequality with Poisson rate function h(u) = (1 + u) log(1 + u) − u, a sharp Bernstein inequality matching the exact Central Limit Theorem Gaussian variance with optimal constant 1/2 via the rational minorant h(u) ≥ u^2 / (2 + 2u/3) on [0, ∞), and optimal Rosenthal-type L^p moment bounds scaling with σ_w √p + c_w ‖w‖_∞ p. (vi) Exact Zero-Variance Annihilation: On rank-k projection DPPs with w = 1, Picard–Lindelöf uniqueness guarantees K_λ ≡ K is an invariant stationary trajectory of the Riccati flow, V(λ) ≡ 0, and the Bennett bound vanishes identically for all t > 0, recovering deterministic conservation. We generalize our results to continuous DPPs on Polish spaces via direct L^2(E, ν) trace-class operator calculus and normalized Galerkin projections, establish matrix Bernstein concentration bounds for symmetric matrix statistics via an ε-net sphere covering (prefactor 2 · 9^d) and commuting simultaneous diagonalization (prefactor 2d), and reveal an intrinsic connection to Nesterov–Nemirovski self-concordance. Applications to uniform spanning trees, randomized numerical linear algebra, and Ginibre eigenvalue fluctuations demonstrate exponential improvements over classical bounds. MSC 2020 Subject Classification: Primary 60G55, 60E15; Secondary 15A45, 60F10, 47B10.