Let d≥ 1, let E_d be the set of unoriented nearest-neighbor edges of Z^d, and let omega=(omega_e)_e∈ E_d∈(0,∞)^E_d be a translation-invariant ergodic random environment. Assume E[omega_e]<∞ and E[omega_e^-1]<∞ for a representative edge e. For the continuous-time random walk with generator (L^omega f)(x)=∑_y:,| y-x|=1omega_x,ybigl(f(y)-f(x)bigr), does there exist a deterministic covariance matrix Sigma such that, for P-almost every environment omega, the laws of bigl(varepsilon X_t/varepsilon^2bigr)_t≥ 0 under the quenched law P^omega_x converge, as varepsilon downarrow0, to the law of a Brownian motion with covariance matrix Sigma?

Random-conductance quenched FCLT at the minimal-moment threshold

Let $d\ge 1$ , let $E_d$ be the set of unoriented nearest-neighbor edges of $\mathbb{Z}^d$ , and let $\omega=(\omega_e)_{e\in E_d}\in(0,\infty)^{E_d}$ be a translation-invariant ergodic random environment. Assume $\mathbb{E}[\omega_e]<\infty$ and $\mathbb{E}[\omega_e^{-1}]<\infty$ for a representative edge $e$ . For the continuous-time random walk with generator

(L^\omega f)(x)=\sum_{y:\,\lvert y-x\rvert=1}\omega_{\{x,y\}}\bigl(f(y)-f(x)\bigr),

does there exist a deterministic covariance matrix $\Sigma$ such that, for $\mathbb{P}$ -almost every environment $\omega$ , the laws of $\bigl(\varepsilon X_{t/\varepsilon^2}\bigr)_{t\ge 0}$ under the quenched law $P^\omega_x$ converge, as $\varepsilon\downarrow0$ , to the law of a Brownian motion with covariance matrix $\Sigma$ ?

References

Primary source

arXiv

Additional references

Quenched functional central limit theorem for the random conductance model under minimal moments — arXiv — Leonid Kolesnikov, Yannic Steenbeck

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle the random-conductance limit theorem at the weakest finite-moment threshold, but the result has not been independently checked.

The problem asks whether the quenched functional central limit theorem holds for strictly positive stationary ergodic nearest-neighbor conductances assuming only a finite first moment. Earlier literature identified this as open; a new preprint claims the affirmative result at that threshold.

Known results

Biskup: dimension $2$ under the stronger condition $\mathbb{E}(\mu_e\wedge\mu_e^{-1})<\infty$ .
Andres, Deuschel, and Slowik: QFCLT under $\mathbb{E}\mu_e^p<\infty$ and $\mathbb{E}\mu_e^{-q}<\infty$ with $p^{-1}+q^{-1}<2/d$ .
A 2019 result established a quenched invariance principle under moment conditions on conductances and inverses, but not under only a finite first moment.

September 2026 claimed resolution

Leonid Kolesnikov and Yannic Steenbeck's preprint, Quenched functional central limit theorem for the random conductance model under minimal moments, claims the theorem in the critical regime $\mathbb{E}\mu_e<\infty$ for strictly positive ergodic conductances. The claim is not independently confirmed or peer reviewed.

Current status (as of September 2026): A preprint claims the finite-first-moment case is solved, but that claim remains unverified, so the problem is not independently settled.

Sources

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