Optimal Cumulant Inequalities and Exact-Variance Bennett Concentration for Strong Rayleigh Measures and Sector-Free Polynomials

This paper establishes sharp all-order cumulant inequalities and leading-constant-1, exact-variance Bennett and Bernstein concentration inequalities for sector-free polynomials and general Strong Rayleigh probability measures. First, for any non-monomial polynomial with non-negative coefficients that is zero-free in an open sector of half-angle theta around the positive real axis, we prove an exact positive integral representation for its logarithmic derivative and exponentially tilted variance against a unique finite Borel measure on the real line. The proof combines a Poisson-Jensen midline identity on the double-width strip with an explicit hyperbolic-tangent subtraction of boundary poles. In particular, the tilted variance is shown to be an exact positive convolution of the scaled single-trial Bernoulli variance kernel. This structural representation yields exact angular monotonicity across the zero-free sector, strict log-concavity on the closed half-width strip, and all-order cumulant-to-variance bounds governed by the extremal cumulant ratios of a single Bernoulli trial, with optimal constants attained whenever pi/theta is a positive integer. Second, combining this univariate representation with real-stable sector-freeness, rational weight approximation, and symmetric homogenization, we resolve the optimal cumulant and Bennett concentration problem for arbitrary real-weighted linear statistics of general Strong Rayleigh measures. We show that the cumulant generating function is generalized self-concordant—with the third cumulant strictly bounded by the exact ensemble variance times the natural weight-oscillation scale—and derive two-sided, constant-1 Bennett, Bernstein, and Rosenthal-type moment inequalities governed by the exact variance rather than independent variance proxies. These bounds apply unconditionally to all k-determinantal point processes (k-DPPs), discrete and continuous self-adjoint determinantal point processes on Polish spaces, conditional Bernoulli (Hajek rejective) sampling, sampling without replacement, and non-determinantal Strong Rayleigh laws such as external-field basis measures of the Vamos matroid. We also prove tail-level optimality of the weight scale and show by explicit counterexample that the results fail under strict pairwise negative correlation alone. Finally, we investigate a strengthened multivariate complex log-modulus convexity property for real-stable and hyperbolic polynomials. We establish this property for Hermitian determinantal pencils with independent exponents via block Schur complements, for all 2-homogeneous and (n-2)-homogeneous Strong Rayleigh measures via Pauli-Clifford representations, and for conditional Bernoulli laws via two-copy trigonometric positivity, while constructing an explicit univariate real-rooted exponential pencil counterexample that motivates the general Hyperbolic Pencil Conjecture. MSC 2020 Subject Classifications: Primary 60E15, 60C05, 60G55; Secondary 30C15, 15A15, 26C10, 62D05.