Intrinsic Gaussian Tensor Geometry and Random Operator Spectra
arXiv:2606.29292
The paper introduces a covariance‑profile technique to build Schatten‑class multiplication operators from Wick powers of Gaussian fields on smooth closed Riemannian manifolds, providing moment estimates, convergence results, and sharp summability conditions.
Abstract
We develop a deterministic coefficient calculus for operator-valued Gaussian chaoses and connect it to spectral information for skew random operators. At each homogeneous order, the oriented input--output flattenings form an intrinsic Schatten-scale coefficient space: a quotient sum below exponent , the Hilbert tensor space at , and a compatible intersection above . Decoupled Gaussian randomization and same-field Wiener--Itô evaluation are two-sided on this scale, so coefficient convergence and random-operator convergence are equivalent. Global coefficients are reconstructed from local blocks by order-continuous reconstruction lattices. Composing this deterministic step with exterior algebra yields a tensor-to-spectrum theorem: for Pfaffian-compatible even towers, the exterior norms recover the Fredholm product and the complete nonzero skew spectrum, with dimension-free weak-Schatten bounds from local profile data. Chronological Itô tensors and operational covariance spectra provide canonical stochastic realizations. On closed Riemannian manifolds, covariance--Gram estimates produce canonical Schatten-class Wick multiplication operators, including the full finite Schatten range and a sharp logarithmic construction at the critical smoothing threshold.
85 pages