AI 中文总结
本文针对复Ginibre对数气体的实对称可观测量建立最优Poincaré不等式，通过结合Vandermonde变换等方法完成证明，确定了对应过阻尼Langevin动力学的精确谱隙。
AI 中文摘要
我们为复Ginibre对数气体的实值对称可观测量建立了最优Poincaré不等式，质心可观测量的实部和虚部可达到等号。等价地，我们确定了实对称可观测量对应的过阻尼Langevin动力学的精确谱隙。该对数气体能量的Hessian下方无界，无法直接应用标准凸性论证，因此我们结合Vandermonde变换、全纯投影以及对应Hörmander-Berndtsson估计常曲率情形的复高斯d-bar谱隙估计完成证明。
英文摘要
We establish an optimal Poincaré inequality for real-valued symmetric observables of the complex Ginibre log-gas. Equality is attained by the real and imaginary parts of the center-of-mass observable. Equivalently, we determine the exact spectral gap of the associated overdamped Langevin dynamics, for real symmetric observables. The Hessian of the energy of this log-gas is unbounded below, so standard convexity arguments do not directly apply. The proof instead combines a Vandermonde transform, a holomorphic projection, and a complex Gaussian d-bar spectral-gap estimate, corresponding to the constant-curvature case of the Hörmander-Berndtsson estimate. It is short and self-contained. It remains valid, beyond the quadratic potential, for rotationally invariant additive plurisubharmonic potential perturbations. Additionally, we provide four alternative proofs, two of which yield sum-of-squares formulas for the deficit, based respectively on a Hermite expansion and on an integrated Bochner-Kodaira formula, the other two use the spectral analysis of a two-sided number-operator factorization and a Hermite-Slater polynomial expansion. We furthermore present new results related to linear statistics, Gaussian factorization, polynomial eigenfunctions, log-Sobolev inequalities, matrix lift and eigenvector overlaps, and non-quadratic potentials.
Comments30 pages, 2 graphics
