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arXiv · 2610.16321

Optimal information loss for Liouville spectral responses

Chunhao Cai
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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 $γ^2$ 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 $L=-(2π)^{-1}Δ_D$ on the unit disk $D$. A quadratic correction of the readouts attains the same fixed-design loss to second order.

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Chunhao Cai. 2026-10-12. Optimal information loss for Liouville spectral responses. https://arxiv.org/abs/2610.16321

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