Linear Response for Stationary Random Fields

Definitions and mathematical statement

Let the intended stationary random field have a joint law invariant under spatial translations. For the macroscopic test observable in the report, write $W_r=Z_0(h_r)$, where the subscript denotes centering, so ${\mathbb{E}} W_r=0$. The field's index space and state space, the statistic $Z_0$, test-function class, and rescaling $h\mapsto h_r$ are not defined in the extraction.

The first request is for a moderate-deviation principle under sufficiently general dependence and integrability hypotheses, described only by examples such as exponential decay of correlations and finite exponential moments. The stronger requested linear-response asymptotic is exactly \[ \frac{1}{r^2\varepsilon^2}\log{\mathbb{E}} \exp\left(\varepsilon r^2 Z_0(h_r)\right) \longrightarrow\frac{\sigma^2}{2}\|\Delta h\|^2 \] as \[ r\to\infty,\qquad \varepsilon(\log r)^2\to0. \] The expression requires $\varepsilon\ne0$. In a joint-limit reading, it is to hold along every admissible sequence of pairs satisfying these two conditions, with the intended sign/range of $\varepsilon$ and admissibility of $h$ supplied by the report. The displayed source does not impose $r|\varepsilon|\to\infty$, and that extra condition has not been inserted into the stronger linear-response assertion. The Laplacian, norm, integration measure, and variance parameter $\sigma^2$ are not specified; in particular, $\|\Delta h\|$ must not be replaced by $\|h\|$.

For clarity, a scalar moderate-deviation statement needs random variables $Y_r$, speeds $a_r\to\infty$, and a rate function $I$, with the large-deviation lower bound on open sets and upper bound on closed sets at speed $a_r$. In a compatible quadratic scenario suggested by the displayed formula one would take \[ a_r=r^2\varepsilon_r^2,\qquad Y_r=W_r/\varepsilon_r, \] requiring additionally $r|\varepsilon_r|\to\infty$. If the needed signed Laplace-transform limits hold for every scalar tilt, the candidate quadratic rate for $\sigma^2\|\Delta h\|^2>0$ is $I(x)=x^2/(2\sigma^2\|\Delta h\|^2)$. These are explanatory conditional choices, not supplied definitions or a proved moderate-deviation theorem. Degenerate variance requires separate treatment.

The record gives no quantitative mixing/cumulant estimates, uniform exponential-integrability conditions for the observables, or topology for a possible field-valued principle. Two-point correlation decay and one-site exponential moments alone are not silently interpreted as every dependence estimate required for the requested conclusion. The supplied assessment dated 2026-08-21 reports model-dependent theorems, not a theorem at the extraction's generality.

Short English statement

The problem asks for general conditions ensuring moderate deviations and an asymptotically quadratic logarithmic moment-generating function for centered macroscopic random-field statistics. The exact scaling and joint limit are retained, while the field, observable, norm, and quantitative dependence assumptions remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30000965$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Stochastic Analysis (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/25, pp. 1361--1424. https://doi.org/10.4171/owr/2008/25.
• The supplied corrected_verified note reports restoration of the scaling, quadratic limit, and joint regime from the official report. The open label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of scope and current status.