Optimal information loss for Liouville spectral responses
Abstract
We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter γ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals γ2 times a positive compression error for a deterministic second-chaos operator, up to o(γ2), with designs allowed to depend on γ. After enlarging the probe dictionary, the optimal m-response coefficient is asymptotic to |Lg|L1(D)2/(8m), where L=−(2π)−1ΔD on the unit disk D. A quadratic correction of the readouts attains the same fixed-design loss to second order.
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BibTeX
@misc{cai2026optimal,
title = {{Optimal information loss for Liouville spectral responses}},
author = {Chunhao Cai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16321},
eprint = {2610.16321},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16321},
}
RIS
TY - UNPB
TI - Optimal information loss for Liouville spectral responses
AU - Chunhao Cai
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16321
UR - https://arxiv.org/abs/2610.16321
AB - We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter γ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals γ^2 times a positive compression error for a deterministic second-chaos operator, up to o(γ^2), with designs allowed to depend on γ. After enlarging the probe dictionary, the optimal m-response coefficient is asymptotic to |Lg|_L^1(D)^2/(8m), where L=−(2π)^−1Δ_D on the unit disk D. A quadratic correction of the readouts attains the same fixed-design loss to second order.
ER -